MOTOR-21 — Observers and Sensorless Control

FACT — If an electric vehicle cruising in the fast lane at $120\text{ km/h}$ suffers an intermittent resolver cable disconnect, an unprotected FOC inverter instantly loses track of rotor angle—causing sudden total loss of propulsion or dangerous unintended regen drag torque (an ASIL-D safety hazard!). Modern automotive inverters run real-time Sensorless State Observers as software redundancy, allowing seamless transition to Limp-Home Mode so the driver can safely pull over.

In this lesson, we examine state observers, Back-EMF tracking, and zero-speed High-Frequency Injection:

1. Why Sensorless Control? (ASIL-D Limp-Home and BOM Cost)
2. Medium-to-High Speeds: Back-EMF State Estimation
3. Sliding Mode Observer (SMO): High-Gain Robust Tracking
4. Extended Kalman Filter (EKF): Stochastic Nonlinear Filtering
5. The Low-Speed Blindspot: Physical Reality of $E \to 0$
6. Zero-Speed Tracking: High-Frequency Voltage Injection (HFI)
7. Resolving North vs. South (N/S) Magnetic Polarity Ambiguity
8. Hybrid Observer Fusion (HFI $\leftrightarrow$ SMO Transition)

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1. Automotive Motivations for Sensorless Control

Physical position sensors (resolvers, optical encoders) present failure modes: cable chafing, connector corrosion, and mechanical offset drift.

Sensorless algorithms serve two vital roles in electric vehicle architectures:

1. ASIL-D Redundant Limp-Home Capability: When resolver loss-of-signal (LOS) is detected on the highway, control shifts to the software observer in milliseconds. Torque is derated to $30\% - 50\%$, keeping propulsion active to clear traffic safely.
2. Auxiliary Powertrain Cost Reduction: Eliminating physical sensors, harnesses, and RDC converter chips in auxiliary drives (electric water pumps, A/C compressors, radiator blowers).

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2. Back-EMF State Estimation

In the stationary $\alpha-\beta$ frame, stator voltage equations are:

$V_\alpha = R_s i_\alpha + L_s \frac{di_\alpha}{dt} + e_\alpha$

$V_\beta = R_s i_\beta + L_s \frac{di_\beta}{dt} + e_\beta$

Where $e_\alpha$ and $e_\beta$ represent the induced Back-EMF voltages:

$e_\alpha = -\omega_e \psi_{pm} \sin\theta_e$

$e_\beta = \omega_e \psi_{pm} \cos\theta_e$

Extracting $\hat{e}_\alpha$ and $\hat{e}_\beta$ from measured currents and commanded voltages allows direct extraction of rotor electrical angle ($\theta_e$) via four-quadrant Arctan2 or a Phase-Locked Loop (PLL):

$\theta_e = \text{atan2}(-\hat{e}_\alpha, \hat{e}_\beta)$

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3. Sliding Mode Observer (SMO)

The Sliding Mode Observer is the most common automotive observer due to its computational efficiency and robustness against motor parameter variations:

$\frac{d\hat{i}_\alpha}{dt} = -\frac{R_s}{L_s}\hat{i}_\alpha + \frac{1}{L_s} V_\alpha - \frac{1}{L_s} z_\alpha$

$\frac{d\hat{i}_\beta}{dt} = -\frac{R_s}{L_s}\hat{i}_\beta + \frac{1}{L_s} V_\beta - \frac{1}{L_s} z_\beta$

Where $\hat{i}$ is estimated current, and $z$ is the discontinuous sliding control signal:

$z_\alpha = k_{smo} \cdot \text{sgn}(\hat{i}_\alpha - i_\alpha)$

$z_\beta = k_{smo} \cdot \text{sgn}(\hat{i}_\beta - i_\beta)$

When sliding gain $k_{smo}$ is sufficiently high, the observer locks onto the sliding manifold $S = \hat{i} - i = 0$. On the sliding surface, the continuous equivalent control signal directly mirrors the real Back-EMF voltages ($z_\alpha \approx e_\alpha, z_\beta \approx e_\beta$). A low-pass filter (LPF) removes switching chattering, with software phase delay compensation.

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4. Extended Kalman Filter (EKF)

For demanding applications with high phase current harmonic noise, the Extended Kalman Filter (EKF) provides optimal minimum-variance stochastic estimation.

The state vector is defined as $x = [i_\alpha, i_\beta, \omega_e, \theta_e]^T$.

• Prediction: Propagates nonlinear machine dynamics via Jacobian state transition matrices ($F$), projecting covariance ahead.
• Correction: Weights current innovation errors via Kalman Gain ($K$), correcting rotor angle in real time.

EKF provides superior noise immunity and dynamic filtering, but consumes roughly $4-5\times$ more CPU instruction cycles than SMO.

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5. The Low-Speed Blindspot ($E \to 0$)

All Back-EMF observers share an absolute physical limitation:

$e \propto \omega_e \cdot \psi_{pm}$

• At standstill or very low rotational speeds ($< 5\% - 10\%\ n_{base}$), Back-EMF voltage is virtually zero.
• The microvolt-level EMF is completely submerged beneath inverter dead-time distortion and ADC quantization noise floor ($SNR \to 0$).
• Consequently, Back-EMF observers cannot determine rotor angle at standstill or low speed.

Launching a 2.5-ton EV uphill requires full standstill launch torque. Standstill position detection requires an entirely different physical phenomenon: magnetic saliency.

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6. Zero-Speed Solution: High-Frequency Injection (HFI)

In IPMSM and SynRM machines, magnetic saliency produces distinct directional inductances ($L_d \ne L_q$).

High-Frequency Injection (HFI) injects a non-torque-producing high-frequency AC carrier voltage ($500\text{ Hz} - 2000\text{ Hz}$) into the stator:

$V_{hfi} = V_h \cos(\omega_h t)$

The high-frequency stator current response modulates as a function of rotor orientation:

$i_{hfi} \propto \frac{V_h}{\omega_h} \cdot \left[ \frac{L_d + L_q}{2 L_d L_q} - \frac{L_q - L_d}{2 L_d L_q} \cos(2\theta_e) \right]$

Demodulating this high-frequency current extracts rotor angle $\theta_e$ without any physical shaft motion!

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7. Magnetic Polarity Identification (North vs. South)

Because saliency tracking relies on a $\cos(2\theta_e)$ term, it yields an orientation axis with $180^\circ$ ambiguity: it identifies the magnetic line, but cannot distinguish the North (N) pole from the South (S) pole.

If stator current is applied with $180^\circ$ error, the vehicle jerks violently in reverse instead of forward.

Polarity Detection via Magnetic Saturation:

Stator laminations exhibit nonlinear B-H saturation:

1. The iron core facing the magnet North pole is already partially pre-saturated by permanent magnet flux.
2. A positive test pulse ($+d$) drives the core deeper into saturation; inductance collapses and current rises steeply.
3. A negative test pulse ($-d$) un-saturates the core; inductance rises and current ramps up slower.
4. Comparing the peak current amplitudes conclusively identifies the North pole in under $2\text{ ms}$ before inverter gating begins.

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8. Hybrid Observer Handover (HFI $\leftrightarrow$ SMO)

Automotive sensorless drives seamlessly transition between algorithms across the vehicle speed envelope:

1. $0 - 300\text{ rpm}$: Pure HFI active. Enables standstill hill launches and creep mode.
2. $300 - 600\text{ rpm}$: Blended transition region. HFI carrier is smoothly attenuated while SMO output gains weight.
3. $> 600\text{ rpm}$: Pure SMO / EKF active. HFI is shut off to eliminate acoustic carrier whine and unnecessary core loss, delivering smooth operation up to maximum vehicle speed.

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9. Summary

• Sensorless FOC provides essential ISO 26262 ASIL-D Limp-Home redundancy for primary traction and reduces BOM cost in auxiliary drives.
• SMO and EKF track Back-EMF to estimate rotor position accurately at medium and high speeds.
• Back-EMF collapse at standstill requires High-Frequency Injection (HFI) leveraging rotor magnetic saliency ($L_d \ne L_q$).
• Magnetic saturation pulses resolve the $180^\circ$ North/South ambiguity.
• Seamless hybrid handover between HFI and SMO ensures robust torque production from zero to maximum motor speed.

Next Lesson

• MOTOR-22 — Electromagnetic Design Principles: Slot/pole combinations, stator tooth skewing, cogging torque minimization, and electrical steel selection.