PWM-Based Back-EMF Extraction

• PWM-based Back-EMF extraction is a technique that utilizes high-frequency PWM switching signals to recover induced voltage signals for sensorless control in electric machines.
• It employs methods such as synchronous demodulation and low-pass filtering to isolate back-EMF from PWM noise during precise sampling intervals.
• This approach minimizes hardware needs while ensuring reliable rotor position and speed estimation even at low speeds or transient events.

Pulse-width modulation (PWM)-based back-electromotive force (back-EMF) extraction refers to techniques that exploit the intrinsic high-frequency switching of PWM-driven inverters to recover electrically induced voltage signals (back-EMF) in electrical machines and actuators. These methods allow for sensorless estimation of rotor position, velocity, or resonant frequency in systems such as brushless motors and resonant electromagnetic actuators. By leveraging the naturally injected high-frequency content created by the PWM carrier, back-EMF extraction can be achieved with minimal additional hardware, enabling robust control and estimation even down to zero speed or during transient resonant events.

1. Physical and Mathematical Foundations

The fundamental principle behind PWM-based back-EMF extraction is that the electrical drive’s switching action injects a high-frequency disturbance into the system, which can be leveraged to excite observable electrical or electromechanical responses. In the context of a surface-mounted PMSM under PWM excitation, the stator voltage equations in the rotating d–q frame include an explicit “injection” term derived from PWM carrier ripple:

$$v_s^{dq}(t) = R_s\,i_s^{dq}(t) + L\,\frac{d\,i_s^{dq}}{dt}(t) + n\omega(t)\,L\,J\,i_s^{dq}(t) + v_{\mathrm{inj}}^{dq}(t)$$

where $v_{\mathrm{inj}}^{dq}(t)$ is a fast-varying component inherited from the PWM switching. This naturally introduces high-frequency excitation into the system without requiring a separate injection signal, as shown in sensorless rotor position methods for both PMSM and BLDC configurations (Surroop et al., 2020, Gamazo-Real et al., 2024).

PWM-driven phase currents reveal a small yet measurable current ripple, induced by the high-frequency components of the inverter output voltage. In single-coil systems (such as linear resonant actuators), the terminal voltage under PWM drive contains $e_{\mathrm{bemf}}(t)$ as defined in

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

and direct back-EMF sampling is feasible during strategically chosen switching intervals (Cho, 2018).

2. Demodulation and Signal-Processing Methodology

Extraction of the back-EMF signal requires demodulating the high-frequency “virtual measurement” component from current or voltage signals. The methodology includes the following:

• Synchronous demodulation: Multiplying the measured current ripple by a “template” function $s_1$ (the zero-mean primitive of the PWM carrier ripple function) and integrating over the PWM period isolates the back-EMF-associated component,

$$y_v(t) := \frac{1}{T_{\mathrm{pwm}}} \int_{t-T_{\mathrm{pwm}}}^{t} i_s^{\alpha\beta}(\tau)\;s_1\!\big(u_s(\tau),\,\tfrac{\tau}{T_{\mathrm{pwm}}}\big) \, d\tau$$

This recovers a quantity algebraically linked to the rotor position, particularly via the machine saliency matrix $S(\theta)$ and the “injection gain” $A^{\alpha\beta}$ computed from known voltages (Surroop et al., 2020).

• Moving average or low-pass filtering: When sampling the floating-phase terminal voltage during PWM “off” intervals, the extracted voltage $\hat{e}_i(t)$ contains switching noise and high-frequency harmonics. Digital filtering (IIR or FIR, tuned near 500–1 000 rad/s) yields the fundamental back-EMF waveform suitable for zero-crossing detection (Gamazo-Real et al., 2024, Cho, 2018).
• Subtractive reconstruction: In resonant actuators, back-EMF is computed using

$$e_{\mathrm{bemf}}(t) = u(t) - R i(t) - L \frac{di(t)}{dt}$$

during 'off' windows, where $v_{\mathrm{inj}}^{dq}(t)$0 and $v_{\mathrm{inj}}^{dq}(t)$1 are negligible, direct measurement of the terminal voltage yields the back-EMF (Cho, 2018).

3. System Architectures and Implementation Strategies

PWM-based back-EMF extraction is realized in several system architectures:

• Sensorless PMSM and BLDC control: Estimation is performed by strategically sampling terminal voltages or phase currents in synchrony with the PWM carrier. For BLDC motors under six-step commutation, one phase is left floating in each sector; during the corresponding off-interval, the floating-phase terminal voltage equals the phase back-EMF (plus a diode drop) (Gamazo-Real et al., 2024). In vector-controlled PMSMs, side-band ripple extraction and demodulation techniques reconstruct salient rotor-position-dependent signals (Surroop et al., 2020, Surroop et al., 2020).
• Resonant electromagnetic actuators: Alternating pulse protocols (PWM “on”-“off” sequencing) excite and measure the system’s mechanically induced back-EMF, enabling real-time resonance tracking, amplitude adaptation, and residual-vibration cancellation (Cho, 2018).

A high-level algorithmic flow across these methods typically includes:

1. Apply the PWM signal or “on”-“off” sequence to the plant.
2. Measure the terminal quantities (current or voltage) at precise, precomputed sampling instants.
3. Demodulate and filter to isolate the small ripple or back-EMF-related signal.
4. Estimate position, speed, or frequency from the processed signals.
5. Feed this information back for commutation, resonance tracking, or vibration suppression.

4. Performance Considerations and Practical Constraints

Empirical results consistently confirm that PWM-based back-EMF extraction delivers accurate sensorless estimation and tracking, provided implementation is carefully synchronized to PWM carrier timing and appropriate signal processing is applied:

• Position reconstruction in PMSMs via least-squares demodulation of PWM-induced currents achieved RMS errors of $v_{\mathrm{inj}}^{dq}(t)$2 rad and maximum steady-state errors $v_{\mathrm{inj}}^{dq}(t)$3 rad at low speeds (5 Hz), with further reduction to $v_{\mathrm{inj}}^{dq}(t)$4 rad using interleaved carriers (Surroop et al., 2020).
• BLDC back-EMF extraction through PWM-synchronous floating-phase sampling delivers position accuracy of $v_{\mathrm{inj}}^{dq}(t)$5–$v_{\mathrm{inj}}^{dq}(t)$6 electrical degrees at medium to high speeds, with robust operation from $v_{\mathrm{inj}}^{dq}(t)$7 times rated speed to above $v_{\mathrm{inj}}^{dq}(t)$8 times rated speed (Gamazo-Real et al., 2024).
• Resonant actuator tracking using this approach reduces residual vibration by over 90% compared to open-loop actuation, with resonant-frequency tracking accuracy within $v_{\mathrm{inj}}^{dq}(t)$90.3% of the nominal value (Cho, 2018).

Key practical constraints include:

• Noise susceptibility and dead-time: Measurement noise and inverter dead-time effects can contaminate demodulation; filtering around the PWM band is standard (Surroop et al., 2020, Gamazo-Real et al., 2024).
• Sampling synchronization: Precise selection of sampling instants post-switching is required to avoid switching transients and guarantee accurate back-EMF capture (Cho, 2018, Gamazo-Real et al., 2024).
• Low-speed (or zero-speed) operation: In PMSM, observability persists down to standstill if machine saliency ($e_{\mathrm{bemf}}(t)$0) exists. For BLDC, open-loop starting is required before reliable back-EMF extraction is achieved (Gamazo-Real et al., 2024).
• Parameter sensitivity: Compensation for armature resistance, inductance, and parasitic diode drops is necessary for quantitative back-EMF estimation at all speeds.

5. Comparative Features and Application Domains

PWM-based back-EMF extraction demonstrates the following advantages compared to classical signal-injection or sensor-reliant approaches:

Feature	PWM-Based Extraction	Dedicated Injection/External Sensor
Hardware Overhead	Minimal, leverages existing PWM inverter	Additional circuitry or sensors needed
Observability (low speed)	Achievable with machine saliency or resonance	Dependent on injection amplitude
Implementation	Synchronized sampling, digital filtering	Extra signal generation, filtering
Robustness	Sensitive to PWM/jitter, manageable by design	External noise or drift possible

Application domains include sensorless field-oriented control of PMSMs and BLDCs for automotive, industrial, and robotics sectors, as well as compact haptic actuators in mobile devices (Surroop et al., 2020, Gamazo-Real et al., 2024, Cho, 2018). Sensorless resonance tracking in LRA motors, feedforward amplitude regulation, and residual-vibration shaping in actuators require no additional velocity or displacement sensors, simplifying controller architectures for low-cost systems (Cho, 2018).

6. Generalization and Extensions

PWM-based back-EMF extraction arises as a generalization of classical signal-injection frameworks, substituting explicit external probing with the internal PWM carrier and its resulting harmonics (Surroop et al., 2020, Surroop et al., 2020). The theoretical underpinning relies on second-order averaging: the plant response is decomposed into slow and fast components, with the latter serving as a vehicle for the “virtual measurement” of otherwise inaccessible quantities (such as back-EMF or its derivative). The demodulation algorithms are systematic and extend across a class of affine-input systems.

Further generalizations include:

• Use of adaptive filtering bandwidths and model-based refinements to handle parameter uncertainties and broaden speed range (Gamazo-Real et al., 2024).
• Injection of synthetic HF signals in special cases to resolve observability at standstill in non-salient machines (though this was not required in the PMSM cases studied).
• Integration with observer design, such as sliding-mode observers or Kalman filters, to further enhance estimation robustness in noisy or time-varying environments (Gamazo-Real et al., 2024).

7. Limitations and Possible Improvements

Despite high demonstrated accuracy, PWM-based back-EMF extraction faces intrinsic limitations:

• Extraction bandwidth is limited by the PWM frequency; only one reliable update per cycle is available (Surroop et al., 2020).
• The necessity for precise parameter knowledge ($e_{\mathrm{bemf}}(t)$1, $e_{\mathrm{bemf}}(t)$2, diode drop) for high-fidelity absolute back-EMF estimation (Gamazo-Real et al., 2024).
• Degradation of position or speed estimation accuracy near zero-crossings due to filter group delay.
• Open-loop starting requirements and loss of information at extreme low speeds for BLDC drives without saliency (Gamazo-Real et al., 2024).

Improvements discussed include adaptive selection of filter bandwidth, machine learning integration for parameter adaptation, and use of high-resolution timers for reduced detection latency. Software compensation for hardware offsets and digital filtering enhancements help maintain estimation quality across a wide operational range.

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PWM-based back-EMF extraction constitutes a robust, hardware-efficient, and theoretically principled method for sensorless estimation in power electronics and electromechanical systems. Its reliance on intrinsic PWM injection harmonizes observability and implementation simplicity, underpinning modern sensorless control algorithms for rotating machines and precision actuators (Surroop et al., 2020, Cho, 2018, Surroop et al., 2020, Gamazo-Real et al., 2024).