Online Error Calibration of Rotor Position Sensor for Automotive Electric Actuator Based on Permanent Magnet Synchronous Motor

2.1 Position Sensor Errors

Axial-mounted position sensors calculate the rotor position using the orthogonal flux component of the sensor magnet attached to the rotor. In the ideal case, the measured flux is expressed as follows.

Here, the estimated position at this time is as follows.

In this case, the magnetic flux trajectory is circular and the estimated rotor position matches the actual position, so no error occurs. However, in Hall sensor-based position detection, various measurement errors can occur due to mechanical alignment errors, the most common cause of which is the eccentricity of the position sensor.⁸⁾

Fig. 3

Position sensor structures: (a) normal state, (b) eccentric state

Eccentricity refers to a state in which the rotation center of the sensor magnet attached to the end of the rotation axis and the measurement center of the position sensor do not coincide and are mechanically misaligned. This eccentricity occurs due to manufacturing tolerances, assembly errors, etc. and inevitably occurs in actual systems.

If position sensor eccentricity exists, the magnetic field distribution recognized by the sensor will differ from the ideal case, resulting in distortion of the position signal. In particular, the ideal sin/cos signal contains additional harmonic components, which lead to position estimation. This distortion is not a simple offset or scale change, but the following types of errors occur.

Here, k represents the amplitude of the error, and ϕ represents the phase offset of the error. Fig. 4 shows the position error waveform caused by this eccentricity. The position errors due to eccentricity manifest including periodic harmonic ripple dominated by phase offset and second harmonics.

Fig. 4

Angle measurement error waveform due to eccentricity

Such errors directly affect the control performance of position sensor-based PMSM drive systems. If errors are included in the position sensor values, the entire current control loop is affected, resulting in degraded control performance and increased torque ripple.^9,10)

2.2 Offline Correction of Position Sensor Errors

As shown in Fig. 5, the offline correction method for compensating for position sensor errors involves using the driving motor of a dynamometer to rotate the target motor at a constant speed and obtaining an absolute reference angle through a high-precision encoder. The error data for each angle derived from this is then constructed into a lookup table to correct the position sensor error in real time during operation. This method requires the construction of a separate, complex testbed in the mass production process, which not only significantly increases manufacturing costs and inspection time but also has limitations in that errors cannot be recorrected on-site if they fluctuate due to temperature changes or mechanical wear after the motor is installed in an actual vehicle or industrial site.

Fig. 5

Conventional position sensor error measurement

3.1 Online Compensation of Position Sensor Offset Errors

To compensate for the phase offset caused by mechanical assembly errors during the installation of the position sensor, a zero-current control technique during the coasting state is employed.¹¹⁾

In a general synchronous rotation d-q coordinate system, the PMSM voltage equations are expressed as follows:

When the motor is operated in the coasting state and controlled such that i_d = 0 and i_q = 0 (zero-current control), the voltage equations are ideally simplified as follows:

In other words, in an ideal alignment state where there is no phase offset in the sensor, the d-axis voltage becomes 0, and only the back EMF proportional to the rotation speed of the motor should appear in the q-axis voltage. Assuming that there is an offset of Δθ between the actual rotor position θ and the position θ_s measured by the position sensor due to the mechanical assembly error of the position sensor, the relationship is as follows:¹²⁾

Fig. 6

Position sensor offset measurement

Due to this phase offset Δθ, in the estimated coordinate system recognized by the controller, the back EMF component that actually exists only on the q-axis is projected onto the d-axis and q-axis, respectively.

Using the d-q axis voltage measured during zero current control, the sensor's offset Δθ can be directly calculated through the arctangent function as follows:

By applying the estimated Δθ as a compensation value to the position detection logic, the controller can perform precise torque control based on an accurate d-q coordinate system.

3.2 Online Compensation Method for Position Sensor Harmonic Errors

The online compensation of position sensor harmonic errors is a software-based technique that automatically compensates for position sensor errors using only the control and estimation algorithms of the motor drive controller, without requiring an additional motor or a high-precision encoder. The virtual reference angle is estimated using the three-phase back EMF. The detected back EMF is converted to a rectangular coordinate system (Clarke Transform) and then an arctangent operation is performed to derive the reference angle.

Fig. 7

Bloc diagram of error compensation table

3.2.2 ANF Design Considering Variable Frequency Environments

In situations where the motor driving voltage is applied, it is impossible to directly measure the pure back EMF due to the influence of inverter switching and the applied voltage. Therefore, back EMF measurement must be performed during the deceleration phase where the motor rotates due to inertia.

To eliminate harmonic error components (second harmonics) generated during this measurement process, a notch filter that attenuates only specific frequency bands can generally be utilized. The transfer function G(s) of a typical second harmonic removal notch filter is expressed as follows:¹⁴⁾

$\[ \mathrm{G}\left(\mathrm{s}\right)=\frac{{\mathrm{s}}^2+{\mathrm{\omega }}_{\mathrm{c}}^2}{{\mathrm{s}}^2+2\mathrm{\zeta }{\mathrm{\omega }}_{\mathrm{c}}\mathrm{s}+{\mathrm{\omega }}_{\mathrm{c}}^2} \]$

(20)

Here, ω_c is the center frequency (notch frequency) to be cut off, and ζ is the damping ratio, a parameter that determines the cutoff bandwidth and depth of the filter. As can be seen from the Bode plot in Fig. 8, the notch filter has the characteristic of effectively removing frequency components by rapidly reducing only the gain (magnitude) of the set center frequency band.¹⁵⁾

Fig. 8

Notch filter bode plot

However, during the inertia deceleration phase, the rotational speed of the motor continuously decreases, so the frequency of the second harmonic ripple included in the position angle error also fluctuates. If the center frequency ω_c in the above equation is fixed at a constant, the filter may deviate from the target frequency band as the speed changes, resulting in attenuation of frequency components other than the second harmonic components. Therefore, to effectively accommodate variable frequency environments, it is essential to employ a control structure that adjusts the notch filter center frequency, ω_c, in real time as a function of the motor electrical angular velocity, ω_e, by synchronizing it according to ω_c = 2ω_e.

To implement this approach, an ANF corresponding to frequency changes was used. Fig. 9 shows a block diagram of the proposed variable frequency-responsive ANF.

Fig. 9

Block diagram of ANF-based position estimation

The signal used as the input to the ANF is constructed by receiving the back EMF-based position angle, θ_emf and the filtered and estimated position angle, θ_est as feedback signals. The second harmonic component to be eliminated fluctuates in conjunction with the electrical angular velocity, ω_e. To compensate this component, the estimated fundamental position angle, θ_est is multiplied by 2 to generate two mutually orthogonal variable reference signals, x₁(t) and x₂(t).

$\[ \{\begin{array}{cc}\hfill & x_1\left(t\right)=\sin \left(2{𝜃}_{est}\right)\hfill \\ \hfill & x_2\left(t\right)=\cos \left(2{𝜃}_{est}\right)\hfill \end{array} \]$

(21)

This reference signal generation method performs the same role mathematically as varying the center frequency, ω_c, of the previously mentioned transfer function as a function of 2ω_e. In other words, it ensures that the cutoff frequency of the ANF tracks the actual second harmonic frequency even when the motor decelerates.

Based on the variable reference signals, the Least Mean Square (LMS) algorithm is applied to continuously update two weights $$ {\widehat{W}}_1$$ and $$ {\widehat{W}}_2$$ for estimating the amplitude of the second harmonic component. The weight update is defined by the following integral formula.

Here, μ is a parameter that determines the bandwidth of the ANF, and the PI gain of the Estimator determines the convergence characteristics of the estimated angle. As the value of μ increases, the response speed of the filter improves, but the notch characteristics become smoother, which may degrade harmonic removal performance; conversely, if the value of μ is excessively small, sufficient error compensation cannot be achieved within the deceleration section. Furthermore, if the PI gain is excessively small, the convergence of the estimated angle is delayed, and if it is excessively large, it may track residual harmonic components, leading to increased oscillation of the estimated angle. In this study, μ = 10, K_p = 1000, and K_i = 40000 were applied to simultaneously secure stable error compensation and rapid convergence of the estimated angle during the deceleration section. These parameters were selected by comprehensively considering the convergence speed of the position estimation error and steady-state ripple, and a settling time of approximately 0.05 seconds or less and stable steady-state characteristics were secured.

$\[ \{\begin{array}{cc}\hfill & W_1\left(t\right)=\int 𝜇\cdot {\widehat{𝜃}}_{emf}\cdot \sin \left(2{𝜃}_{est}\right)dt\hfill \\ \hfill & W_2\left(t\right)=\int 𝜇\cdot {\widehat{𝜃}}_{emf}\cdot cos\left(2{𝜃}_{est}\right)dt\hfill \end{array} \]$

(22)

The final output formula is as follows:

$\[ \begin{array}{cccc}\hfill & {\widehat{𝜃}}_{\mathrm{emf} }=\hfill & \hfill & ({\mathrm{\theta }}_{\mathrm{emf}}-{𝜃}_{est })-\{W_1\sin \left(2{𝜃}_{est }\right)+\hfill \\ \hfill & \hfill & \hfill & W_2\cos \left(2{𝜃}_{est }\right)\hfill \end{array} \]$

(23)

Finally, the filtered position change amount $$ {\widehat{𝜃}}_{\mathrm{emf}}$$ derived through ANF passes through the PI controller and integrator inside the subsequent estimator, causing the error component to converge to zero, and as a result, a filtered estimated position angle θ_est with the second harmonic component suppressed can be obtained.

Fig. 10

Verification of tracking performance and convergence speed of the ANF according to the damping ratio

3.3 Overall Structure of the Proposed Error Compensation

The overall control diagram of the PMSM incorporating the previously described adaptive notch filter-based online reference angle estimation technique is shown Fig. 11. The proposed error compensation algorithm has a structure that extracts the sensor's position error during the coasting section to construct a table or derives an error model function through curve fitting. This error information is used for position compensation during normal operation, and the error compensation procedure is as follows:

Fig. 11

Block diagram of the proposed error compensation

First, the offset caused by the mechanical assembly tolerance of the position sensor is corrected using Δθ estimated through zero-current control. The sensor's zero point is initially aligned by subtracting the offset compensation value Δθ from the angle θ_s measured by the sensor.

Second, the three-phase back EMF generated during coasting is transformed into coordinates and an arctangent operation is performed to derive the back EMF-based reference angle. The corresponding angle passes through ANF to eliminate second harmonic error components, generating a high-purity virtual reference angle, $$ {\widehat{𝜃}}_{emf}$$.

Third, the deviation between the zero-aligned sensor angle θ_s and the ANF-filtered back EMF-based reference angle $$ {\widehat{𝜃}}_{emf}$$ is computed to extract the periodic error component θ_err, which arises from factors such as sensor mechanical eccentricity. The extracted error data is mapped to the mechanical position of the rotor and stored within the controller in the form of a table (θ_err Table), or it is converted into a continuous error model function through curve fitting techniques for use.

Finally, during normal operation of the motor, the final control angle θ is derived by subtracting the error value calculated through the data stored in the table or the curve fitting function from the angle acquired by the position sensor.

Photo. 1

Configuration of motor test bench

Photo. 2

Configuration of motor test bench

By applying this position θ, in which both harmonic distortion and offset are compensated, to the speed controller and coordinate converter, current distortion and torque ripple caused by sensor errors can be effectively reduced.

4.1 Experimental environment configuration

An electric motor driving experiment was conducted to verify the proposed adaptive error compensation algorithm. The experimental setup is shown in Fig. 1 and Fig. 2, and the main specifications of the motor used in this experiment and the parameters of the control system are summarized in Table 1 and Table 2 below.

Table 1

Specifications of motor

Table 2

Specifications of experimental system

The position measurement error caused by the assembly eccentricity of the actual sensor magnet and position sensor, which is basically inherent in the manufacturing and mass production process of the test motor, was compensated and the results were verified.

First, to verify the performance of the back EMF-based estimated angle filtered through ANF, the reference angle of the high-precision encoder was compared with the back EMF-based estimated angle. By comparing the estimated angle based on back EMF and the position sensor output value for which precision was secured, the angle error caused by the eccentricity of the position sensor was derived. Afterwards, the position sensor angle was compensated based on the derived error, and the harmonic distortion rate of the phase current waveform was compared to confirm the reduction in position error before and after compensation.

4.2 Experimental results and analysis

In order to reproduce the actual driving environment where back EMF imbalance exists in the experiment, a 3 % magnitude error was applied to the three-phase back EMF, and based on this, the position estimation error before and after applying the algorithm was compared and analyzed.

To simulate the section where the motor rotates and decelerates by inertia after the inverter switching is blocked, the rotation speed of the dynamo side motor coupled with the test motor was set to exponentially decrease according to time t (RPM = 3000 * e^-2t). The results of measured speed response are shown in Fig. 12.

Fig. 12

PMSM Rotational speed (RPM)

The reference angle θ measured by the high-precision encoder on the dynamo side and the estimated angle θ_emf before ANF application, derived from the back EMF applied with a 3 % error are shown in Fig. 13. The results of comparing the angle estimation errors before and after ANF application by plotting the errors (θ − θ_emf) and (θ − θ_est) for the reference angle θ on the same graph are shown in Fig. 14.

Fig. 13

Comparison between the ideal angle and back EMF based angle

Fig. 14

Comparison of position angle error before and after compensation

Through the comparison of position estimation errors in Fig. 13, it can be seen that a second harmonic ripple with an amplitude of 0.03 rad exists due to the back EMF error simulation before ANF application, but the amplitude converges to near zero after ANF application. This confirms that the adaptive mechanism of ANF, which updates the center frequency in real time even under conditions of motor speed fluctuation, attenuates only the second harmonic component caused by back EMF imbalance. Therefore, it can be confirmed that the back EMF-based estimated angle θ_emf compensated through ANF has an estimation accuracy almost identical to that of θ measured by a high-precision encoder and thus possesses a level of reliability capable of replacing the reference angle.

When driving at 2,000 rpm constant speed, Fig. 15(a) shows mechanical and electrical angles, and Fig. 14 (b) shows the position error waveform. Since the number of pole pairs is 5, the electric angle has a frequency that is 5 times faster than the mechanical angle, as confirmed by the period comparison in Fig. 14(a). As a result of analyzing the error components in Fig. 14(b) by filtering out high-frequency components using a low-pass filter, errors in the form of second harmonics relative to the mechanical angle were predominantly observed. This causes the electrical 5th and 7th harmonic components inherent in the three-phase drive system to be strongly amplified due to the influence of the 2nd/5th low-frequency electric angle in a system with 5 pole pairs.

Fig. 15

(a) Mechanical angle and electrical angle, (b) angle error and low-pass filtered error

The position error data obtained through the experiment is as shown in Fig. 16. The error model function formulated using the curve fitting technique based on the obtained data is as shown in Eq. (24).

Fig. 16

Position error table for angle compensation

Fig. 15 is a position error table or error function, and the phase current waveform before and after compensating for the angle error is as shown in Fig. 17.

Fig. 17

Measured phase current waveforms (a) before error compensation, (b) after error compensation

The phase current waveform measured before the application of the error compensation technique is shown in Fig. 16(a), and the phase current waveform measured after the application of the error compensation technique is shown in Fig. 16(b). Since it is difficult to quantitatively verify the reduction in phase current ripple after error compensation, the phase current waveform was analyzed using Fast Fourier Transform (FFT). The analysis results are shown in Fig. 18.

Fig. 18

Comparison of phase current FFT spectra with and without error compensation

Through the FFT analysis results of the phase current in Fig. 17, it was confirmed that the position error in the form of the 2nd harmonic relative to the mechanical angle manifests as 5th and 7th harmonic components relative to the electrical angle. As a result of applying the proposed error compensation technique to the system to compensate for the position error relative to the mechanical angle, it was verified through actual experiments that the magnitudes of the 5th (833.3 Hz) and 7th (1166.7 Hz) harmonic components included in the phase current relative to the fundamental wave (166.7 Hz) were reduced by more than 5 dB, respectively, compared to before compensation.

To quantitatively evaluate the performance of this proposed technique, the THD was calculated by considering odd harmonic components up to the 17th order relative to the fundamental wave, as shown in the following formula:

The calculated Total Harmonic Distortion (THD) results are shown in Fig. 19. The results showed that the THD before applying the compensation technique was approximately 2.846 %, but after applying the proposed technique, the THD decreased to approximately 1.716 %, confirming that the harmonic distortion of the phase current was significantly reduced.

Fig. 19

Comparison of phase current THD with and without error compensation