Back-EMF Sensing Techniques

• Back-EMF sensing techniques are sensorless methods that exploit induced voltage to estimate rotor position, speed, or vibratory motion in electric drives.
• Techniques include direct voltage sampling, observer-based estimation, and PWM-synchronized extraction, balancing cost, complexity, and noise immunity.
• These methods enhance performance in BLDC, PMSM, and resonant actuators by providing accurate commutation, resonance tracking, and adaptive real-time control.

Back-electromotive force (back-EMF) sensing encompasses a class of sensorless techniques exploiting the voltage generated by electromagnetic induction in machine windings to estimate rotor position, speed, or vibratory motion. These techniques obviate physical position sensors, enabling feedback for commutation, control, and resonance tracking in electric drives and actuators including brushless DC motors (BLDC), permanent-magnet synchronous machines (PMSM), and linear resonant actuators (LRA). The measurable back-EMF, proportional to the rate of change of magnetic flux linkage, is inherently linked to electrical machine dynamics. State-of-the-art methods utilize back-EMF in various model, observer, and switching strategies to achieve high accuracy, noise immunity, and broad speed or frequency coverage. Techniques span direct voltage sampling with filtering, observer-based estimation, and advanced signal injection methods, each with specific advantages and operational constraints (Gamazo-Real et al., 7 Feb 2024).

1. Physical Models of Back-EMF Generation

The generation and measurement of back-EMF are rooted in the electromechanical equations coupling machine current, voltage, and mechanical movement.

In electromagnetic actuators—such as LRAs for vibrotactile feedback—the system is modeled as a mass–spring–damper with force $F(t)=K_j u(t)$ and electrical dynamics

$u(t)=R i(t) + L \frac{di}{dt} + E_b(t)$

with the back-EMF given by $E_b(t) = K_b \dot{x}(t)$, linking the coil terminal voltage (under open-circuit conditions) directly to the actuator mass velocity (Cho, 2018).

BLDC motors are typically represented by per-phase voltage equations:

$$v_a = R i_a + L \frac{di_a}{dt} + e_a, \quad v_b = R i_b + L \frac{di_b}{dt} + e_b, \quad v_c = R i_c + L \frac{di_c}{dt} + e_c$$

where $e_{a,b,c}$ are the phase back-EMFs, often trapezoidal in shape for BLDCs (Geraee et al., 2018).

For PMSMs in the stationary $\alpha$-$\beta$ frame,

$$u_s = R i_s + \frac{d}{dt}[L i_s] + \omega \varphi J \zeta$$

$\omega$ is the rotor speed, $\varphi$ the flux magnitude, $J$ the canonical $\pi/2$-rotation matrix, and $\zeta$ the flux orientation. The last term, $e_s(t) = \omega \varphi J \zeta$, is the vector back-EMF (Bosso et al., 2020).

The measurement of back-EMF is fundamental in decoupling position and speed information from electromechanical systems without physical sensors.

2. Direct Back-EMF Sensing and Terminal Voltage Methods

Terminal voltage sensing, also known as zero-crossing detection, is prevalent in sensorless BLDC drives. Here, only two windings conduct per interval; the voltage at the floating (non-conducting) terminal reflects both the phase back-EMF and the neutral potential. Back-EMF zero-crossing is detected when the floating terminal voltage equals half the DC link ($v_C \approx V_{DC}/2$), indicating that $e_C=0$. Subsequent commutation is delayed by 30° electrical to optimize torque alignment with the trapezoidal EMF waveform (Gamazo-Real et al., 7 Feb 2024).

This method features:

• Low cost and minimal circuitry requirements
• Applicability from 20% to 100% of rated speed (e.g., 1,000–6,000 rpm)
• Real-time operation facilitated by phase-voltage dividers, low-pass filters (cut-off 1–2 kHz), and comparators

Drawbacks include loss of accuracy at low speed due to diminished EMF amplitude, susceptibility to LPF-induced delays at high speed, and the necessity for open-loop startup prior to closed-loop sensorless operation.

An alternative is third-harmonic voltage integration: summing all phase voltages yields a signal dominated by the 3rd harmonic, which, upon integration, provides rotor position information via its zero-crossings every 60° electrical. This method extends the working range to lower speeds and is robust to noise, at the expense of increased hardware complexity and reliance on machine symmetry (Gamazo-Real et al., 7 Feb 2024).

PWM-based back-EMF extraction further refines sampling by synchronizing measurement with inverter “off” intervals, eliminating LPF delays. Techniques such as virtual neutral elimination and complementary switching enhance sampling fidelity at high speeds or for low-EMF operation, adding system implementation flexibility (Gamazo-Real et al., 7 Feb 2024).

3. Observer-Based and Model-Driven Back-EMF Estimation

Sophisticated estimation approaches exploit mathematical observers designed around machine models. For trapezoidal-EMF BLDCs, sliding-mode observers (SMO) are utilized to estimate back-EMF in a hybrid $(d,q)$ frame under the assumption that back-EMF components remain constant over the PWM period. The observer structure is

$$\dot{\hat{x}} = A\hat{x} + Bu + K\, \mathrm{sat}(y - C\hat{x})$$

where $x=[i_q\ i_d\ e_q\ e_d]^T$, $u=[u_q\ u_d]^T$, and the sliding-mode injection $\mathrm{sat}(\cdot)$ provides robustness to switching noise, eliminating the need for external filtering (Geraee et al., 2018).

Upon convergence, estimates $\hat{e}_q, \hat{e}_d$ yield instantaneous electrical angle via

$$\hat{\theta}_e = \mathrm{atan2}(\hat{e}_q,\,\hat{e}_d)$$

and directly determine commutation events. Gain selection ensures finite-time convergence, often optimized (e.g., via NSGA-II) to balance chattering and tracking latency. The structure is inherently robust, filter-free, and suited for direct torque control applications.

In PMSMs, gradient-descent or immersion and invariance (I&I) observers can estimate both resistance $R$ and the vector back-EMF $h$ by leveraging persistent excitation via high-frequency current injection. The error dynamics for parameter estimation,

$\dot{z} = -\frac{k_z}{L} M(i)z$

with $z$ the parameter estimation error and $M(i)\succeq 0$ the regressor matrix, guarantee exponential convergence on excitation (Bosso et al., 2020). Such observers are integrated with unit circle (Lie group $S^1$) formalisms to robustly estimate position and speed. Attitude observers further refine alignment between estimated and true rotor flux orientation, with provable regional practical asymptotic stability.

4. Specialized Strategies: Resonant Actuators, Signal Injection, and Filtering

In actuators where resonance tracking is essential—such as LRAs for haptics—sensorless control demands the estimation of the (potentially shifting) resonant frequency. Methods described for LRA actuators alternate between drive and high-impedance sense phases, measuring the coil’s induced back-EMF (proportional to the mass velocity) in real time. The application of short voltage pulses excites the system; after the drive, the controller records the back-EMF decay, detects zero-crossings or peaks, and estimates the damped natural frequency:

$\omega_d \approx 2\pi/\Delta t$

using the time between back-EMF peaks, enabling continuous resonance tracking and active residual vibration control. Experimental data demonstrate sub-1 Hz tracking errors and significant reduction (>90%) in residual vibration—critical for crisp vibrotactile feedback (Cho, 2018).

Signal injection methods for PMSM sensorless control involve high-frequency current modulation along magnetically decoupled axes, ensuring observability even near zero mechanical speed. This facilitates robust back-EMF (and resistance) adaptation, with torque tracking unaffected by the injection (Bosso et al., 2020).

In all methods, noise rejection and immunity are central concerns. Boundary-layer saturation in sliding-mode observers, analog/digital low-pass (or band-pass) filters, and judicious PWM timing are standard for suppressing switching artifacts without introducing critical delay or phase distortion.

5. Comparative Performance, Limitations, and Application Mapping

Each back-EMF sensing method presents distinct tradeoffs in speed range, complexity, noise immunity, and implementation cost, summarized in the table below (Gamazo-Real et al., 7 Feb 2024):

Technique	Speed Range	Circuit/Compute Complexity
Terminal Voltage Sensing	~20–100% rated	Minimal (comparators, timer)
Third-Harmonic Voltage Integration	Broad, incl. low	Moderate (summing, integration)
Terminal Current Sensing	≥50% rated	High (multiple comparators)
Back-EMF Integration	Mid/high-speed	Moderate (rectifier, integrator)
PWM-Based Extraction	Near-zero – full	Moderate (timed ADC, logic)

Terminal voltage methods dominate in cost-sensitive, moderate-speed applications (fans, pumps), while PWM-based strategies provide wide speed coverage and minimal analog hardware, making them suitable for automotive, consumer, and battery-powered products. Third-harmonic and integration approaches, while more complex, offer superior performance at low speeds and under severe noise conditions.

All methods share the limitation of open-loop start-up, as back-EMF is absent at standstill. Integrated observer-based schemes and advanced signal injection strategies mitigate observability losses at low speeds (or zero frequency), ensuring full dynamic state reconstruction without mechanical sensors (Bosso et al., 2020).

Experimental and simulation studies report rapid convergence and tracking for observer-based methods, accurate commutation for sliding-mode observers in BLDCs, and substantial improvement in actuator haptic performance with dynamically tracked resonance control (Cho, 2018, Geraee et al., 2018).

6. Advances and Hybrid Approaches

Recent research surveys also address hybrid approaches: model-driven estimation augmented by artificial neural networks, combination of PWM extraction with adaptive observers, and the use of recursive least squares (RLS) for online load and parameter identification (Gamazo-Real et al., 7 Feb 2024, Geraee et al., 2018).

For BLDC and PMSM drives, these hybrid estimators permit adaptation to parameter drift (e.g., resistance variation with temperature), load uncertainty, and non-ideal machine characteristics. Extended Kalman filters, MRAS, and full/pseudo-reduced-order adaptive observers are under active investigation for their ability to generalize and improve robustness across a broad array of machine types and operational domains.

A plausible implication is that, although model complexity and computational burden rise with these hybrid methods, the resultant observability, accuracy, and adaptation justify their deployment in applications demanding stringent dynamic performance.

For LRAs and haptic actuators, simple microcontroller-based back-EMF estimation already achieves real-time resonance tracking and damping without external sensors or prior resonance data, providing robust operation despite manufacturing tolerances, wear, and variable loading (Cho, 2018).

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Modern back-EMF sensing techniques provide a continuum of solutions spanning from minimal circuit, comparator-based designs to mathematically sophisticated observers and adaptive signal-processing architectures. Selection is governed by machine topology, performance requirements, cost constraints, and noise environment. The literature demonstrates that adaptability, robustness, and accuracy of back-EMF-based sensorless control have reached a high degree of maturity for both industrial and consumer applications (Gamazo-Real et al., 7 Feb 2024, Geraee et al., 2018, Cho, 2018, Bosso et al., 2020).