An improved method of initial rotor position detection for PMSM by voltage pulse injection

Abstract

Background

Accurate acquisition of the initial rotor position is crucial for permanent magnet synchronous motor (PMSM) control systems. Starting the motor without initial position detection may lead to excessive current or unexpected reverse rotation. However, the traditional voltage pulse injection method often suffers from insufficient accuracy and magnetic pole misidentification, which reduces the reliability and practical applicability of the detection results.

Objective

To eliminate the magnetic pole misidentification inherent in the traditional voltage pulse injection method and further improve detection accuracy, an improved method based on voltage pulse injection is proposed.

Methods

Firstly, to eliminate the magnetic pole misidentification observed in the traditional method, a magnetic pole identification (MPI) strategy is introduced. The MPI strategy determines whether the detected initial rotor position is consistent with the rotor polarity. If a polarity reversal is identified, the detected position is corrected by adding 180°. Next, to reduce the influence of sampling errors on detection accuracy, the magnetization effect of the injected current is enhanced. The difference among the feedback current responses becomes more pronounced, thereby mitigating the influence of sampling errors. Finally, a curve-fitting method is employed to further improve the detection accuracy. By utilizing the acquired current and position information, the current response curve near the actual rotor position is fitted, and the position corresponding to the peak current is selected as the final detection result.

Results

The effectiveness of the improved method has been validated through experiments conducted on a PMSM drive platform. The experimental results demonstrate that the improved method reduces the position detection error by an average of 32% and eliminates magnetic pole misidentification in the traditional method.

Conclusions

Compared with the traditional voltage pulse injection method, the improved method effectively enhances the accuracy of initial position detection, providing a strong basis for the stable operation of PMSM.

Keywords

Permanent magnet synchronous motor voltage pulse injection initial rotor position magnetic saturation

Introduction

As an important type of motor system, the permanent magnet synchronous motor (PMSM) has been extensively applied in modern industrial manufacturing, automobiles, power electronics, and various other domains due to its reduced dimensions, light weight, superior power concentration, and outstanding dynamic behavior.^1–3

The start-up of the PMSM is a critical step in motor control, and accurate acquisition of the initial rotor position is essential for reliable startup performance. If the initial position is not obtained precisely, the voltage vector is blindly injected into the motor, problems such as overcurrent, rotor locking, and reverse rotation may occur, potentially resulting in serious accidents in practical industrial applications.^4,5 The more accurate the initial position acquisition, the greater the starting torque and the higher the operating efficiency of the motor at the beginning of operation.^6,7 In motor systems equipped with absolute encoders, the initial position can be readily obtained from the encoder signals. However, for systems using incremental encoders or sensorless control strategies, obtaining the initial position is relatively challenging.⁸ The issue of initial position acquisition has attracted considerable attention in recent years.

At present, the mainstream methods for detecting the initial position mainly include the magnetic pole orientation method, high-frequency injection (HFI), and voltage pulse injection.^9–12 The main principle of the magnetic pole orientation is to inject a current vector at a fixed angle, thereby forcing the rotor to align with the specified position and enabling smooth startup. Due to the uncertainty of the motor's initial position, the rotation direction of the motor is not fixed during operation, and is affected by the actual load, the size of the injection current should be adjusted in different cases to prevent the accuracy of the method from being affected. In addition, if the set current vector angle is 180 degrees different from the actual initial position, the electromagnetic torque generated by the motor is almost 0, and the motor rotor will not rotate, which will lead to the 180° identification error. To address this issue, some researchers have proposed a two-step positioning method. After injecting the first current vector, a second current vector that differs by 90 degrees from the first is applied. This ensures that the rotor will move to the angle corresponding to the second current vector, thereby eliminating the 180° error. But the magnetic pole orientation method is not applicable to the condition that the rotation range of the motor cannot be too large or standstill due to the unfixed rotation of the rotor.

The HFI estimates the initial rotor position by injecting a high-frequency voltage signal into the motor and extracting the rotor position information contained in the resulting current response. The acquired signal is processed through filtering stages and subsequently fed into a phase-locked loop (PLL) observer to obtain the initial rotor position. And the position detection can be carried out in the stationary state of the motor. Because HFI requires signal processing steps such as filters, the amount of computation is large, and the design of the filter is also relatively difficult, the application in practice is more complicated.

The main principle of the voltage pulse injection method is to inject a series of voltage vectors with a fixed angle difference into the motor based on the magnetic saturation effect or rotor saliency effect. The corresponding d-axis current generated by each injected voltage vector is sampled, and the rotor position is identified by comparing the magnitudes of the d-axis current responses.¹³ The voltage pulse injection method requires relatively low computational effort and is straightforward to implement. Through continuous iteration, extremely high detection accuracy can be obtained theoretically. However, the voltage pulse injection method has the following several issues. Firstly, the selection of the magnitude of the voltage vector and the injection time is relatively difficult. Secondly, it is sensitive to current sampling accuracy. When the voltage vector is close to the actual angle, the difference among the feedback current responses becomes very small, and the presence of current sampling errors can lead to incorrect judgments, making it challenging to achieve extremely high precision. Additionally, when the effects of current magnetization and demagnetization are not strong, the distinction between the feedback currents corresponding to the N-pole and S-pole positions becomes insufficient, resulting in magnetic pole misidentification. In reference,¹⁴ a detection method that combines q-axis feedback current is proposed to address the issue of insufficient differentiation between d-axis feedback currents in the traditional voltage pulse injection method. After the angular interval is further subdivided, the q-axis current information is utilized to improve the detection accuracy. But, in many existing studies on voltage pulse injection methods, the acquired current information is mainly used for comparison and has not been fully exploited to further enhance detection performance.

Hence, to eliminate magnetic pole misidentification and improve the position detection accuracy, an improved voltage pulse injection method is proposed in this paper. Firstly, magnetic pole identification is introduced to eliminate the detection errors caused by magnetic pole misidentification. Next, the magnetization effect is appropriately enhanced to increase the difference among the feedback current responses as the detection process progresses, thereby reducing the influence of sampling errors. Finally, a curve-fitting method is incorporated to further improve the detection accuracy. By fitting the current response curve near the actual rotor position, the angle corresponding to the peak current is selected as the final detection result. This approach maximizes the utilization of the acquired current information and further enhances the detection performance.

In section II, the theoretical analysis of the voltage pulse injection method is presented. In section III, the specific steps of voltage pulse injection are introduced in detail. The drawbacks of the method are introduced in Section IV. An improved method is explained in section V. The experimental results are presented in Section VI.

Theoretical analysis of the traditional voltage pulse injection method

Establish a virtual $d * q *$ coordinate system, its positional relationship in relation to the actual rotor dq coordinate system and the ABC coordinate system, as shown in Figure 1.

Where $θ_{e}$ is the angular difference between the actual rotor dq coordinate system and the ABC coordinate system, $θ_{c}$ is the angular difference between the virtual $d * q *$ coordinate system and the ABC coordinate system, and $Δ θ$ is the difference between $θ_{e}$ and $θ_{c}$ .

And inject the voltage vector into the virtual dq-axis:

{\begin{matrix} U_{d} * = U \\ U_{q} * = 0 \end{matrix}

(1)

Through the inverse Park transformation, it is converted to the actual rotor dq-axis as follows:

{\begin{matrix} U_{d} = U \cos Δ θ \\ U_{q} = U \sin Δ θ \end{matrix}

(2)

When the motor rotor is in a stationary state ( $w_{e}$ =0), the voltage equations of the dq coordinate system of the PMSM can be expressed as:

{\begin{matrix} U_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} \\ U_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} \end{matrix}

(3)

where

U_{d}

i_{d}

L_{d}

is the stator d-axis voltage, current, and inductance,

U_{q}

i_{q}

L_{q}

is the stator q-axis voltage, current, and inductance in the actual dq coordinate system,

R_{s}

is the stator resistance, and

w_{e}

is the electric angular velocity.

U_{d} *

is virtual

d * - axis

voltage,

U_{q} *

is virtual

q * - axis

voltage.

The feedback current in the actual rotor dq-axis can be obtained from the equation (3):

{\begin{matrix} i_{d} = \frac{U_{d}}{R_{s}} (1 - e^{- (\frac{R_{s}}{L_{d}}) t}) \\ i_{q} = \frac{U_{q}}{R_{s}} (1 - e^{- (\frac{R_{s}}{L_{q}}) t}) \end{matrix}

(4)

Substituting equation (2) into equation (4) and transforming $i_{d}$ , $i_{q}$ to the virtual $d * q *$ axis coordinates through the Park transformation yields:

{\begin{matrix} i_{d} * = \frac{U}{R_{s}} (1 - e^{- (\frac{R_{s}}{L_{q}}) t}) - \frac{U}{R_{s}} E \cos^{2} (Δ θ) \\ i_{q} * = \frac{1}{2} \frac{U}{R_{s}} E \sin (2 Δ θ) \end{matrix}

(5)

E = (e^{- (\frac{R_{s}}{L_{d}}) t} - e^{- (\frac{R_{s}}{L_{q}}) t})

(6)

where

i_{d} *

is virtual

d * - axis

current,

i_{q} *

is virtual

q * - axis

current.

For an interior permanent magnet synchronous motor (IPMSM), the structural characteristic $L_{d} < L_{q}$ results in E < 0. According to the equation (5), the maximum value of $i_{d} *$ can be obtained when $Δ θ$ is 0°. Therefore, the maximum $i_{d} *$ is obtained when the injection voltage vector is aligned with the rotor d-axis. In this way, the initial electrical angle of the motor can be detected by finding the maximum $i_{d} *$ . For surface-mounted permanent magnet synchronous motor (SPMSM), $L_{d} \approx L_{q}$ , resulting in $E \approx 0$ . However, it is possible to utilize the magnetic saturation effect to produce $L_{d} < L_{q}$ .^15–17

Magnetic saturation effect: due to the physical structure limitations of magnetic materials, the magnetic flux does not continuously increase but gradually remains at a saturation level after reaching a certain threshold.

The relationship between the d-axis magnetic flux linkage, inductance, and current is expressed as follows:

ψ_{d} = L_{d} i_{d}

(7)

As shown in Figure 2, as the positive d-axis current increases, $ψ_{d}$ no longer increases proportionally with the current. Instead, when the positive d-axis current reaches a certain level, $ψ_{d}$ gradually approaches saturation. To improve material utilization, the d-axis magnetic circuit of a surface-mounted permanent magnet motor is typically designed to operate at point A in Figure 2, where the magnetic circuit exhibits a certain degree of saturation. When current is applied to the stator windings, the magnetic circuit can easily reach saturation. According to (7) and Figure 2, when $i_{d}$ increases to a certain extent, $ψ_{d}$ approaches saturation and no longer continues to increase. At this point, further increasing $i_{d}$ will instead cause $L_{d}$ to gradually decrease, resulting in $E < 0$ .

After the voltage vector is injected, the positional relationship between the resulting stator flux linkage $ψ_{s}$ and the rotor flux linkage $ψ_{f}$ is illustrated in Figure 3(a) ∼ (d).

When the positional relationship between $ψ_{s}$ and $ψ_{f}$ is as shown in Figure 3(a), the angle between them is 0°. Under this condition, the component of $ψ_{s}$ along the direction of $ψ_{f}$ is maximized, making the d-axis magnetic circuit more prone to saturation, and the magnetic saturation effect reaches its maximum. The injected voltage vector produces a magnetizing effect, and $L_{d}$ attains its minimum value. Meanwhile, $ψ_{s}$ has no component along the q-axis magnetic circuit, thus it does not affect the saturation level of the q-axis magnetic circuit, and $L_{q}$ remains unchanged.

When the positional relationship between $ψ_{s}$ and $ψ_{f}$ is as shown in Figure 3(b), the angle between them is 180°. In this case, the injected voltage vector is opposite to the direction of the rotor d-axis, causing the d-axis magnetic circuit to become desaturated and enter the linear region. The voltage vector thus has a demagnetizing effect, and $L_{d}$ reaches its maximum value. Meanwhile, $ψ_{s}$ has no component along the q-axis magnetic circuit, so it has no effect on the saturation level of the q-axis magnetic circuit, and $L_{q}$ remains unchanged.

When the positional relationship between $ψ_{s}$ and $ψ_{f}$ is as shown in Figure 3(c) and (d), the angle between them is 90° and 270°, respectively. In this case, the injected voltage vector has no component along the rotor d-axis and therefore has no effect on the saturation level of the d-axis magnetic circuit, so $L_{d}$ remains unchanged. Meanwhile, $ψ_{s}$ has a component along the q-axis magnetic circuit. The q-axis magnetic circuit is typically designed to operate at point B in Figure 4, and $ψ_{s}$ generally does not cause saturation of the q-axis magnetic circuit; therefore $L_{q}$ remains unchanged.

When considering magnetic saturation, the variation curves of the d-axis and q-axis inductances with respect to the $Δ θ$ are shown in Figure 5. It can be observed that the d-axis inductance reaches its minimum value at $Δ θ$ = 0° and its maximum value at $Δ θ$ = 180°, while the q-axis inductance is independent of $Δ θ$ .

Therefore, based on the above analysis, for a surface-mounted permanent magnet synchronous motor (SPMSM), by utilizing the magnetic saturation effect, $i_{d} *$ also reach its maximum value when $Δ θ$ = 0°. This indicates that the voltage pulse injection method is also applicable to SPMSM.

Introduction to the voltage pulse injection method

Based on the above principle, the motor's initial position can be detected by injecting a series of voltage vectors with the same angular intervals into the motor stator and then judging the magnitude of the feedback current. To ensure the injection voltage vectors do not affect each other, after each voltage injection, the excitation is interrupted for a short duration to allow the current to return to zero before the subsequent voltage vector is applied. The curve of the injection voltage and its response current change over time is shown in Figure 6.

The definitions of injection time and stop-injection time are given as follows:

T_{o n} = t_{1}

(8)

T_{o f f} = t_{2} - t_{1}

(9)

where

T_{o n}

is the time for voltage vector injection, and

T_{o f f}

is the time for stopping voltage vector injection.

The magnitude of the injection voltage vector $U_{m}$ and the injection time $T_{o n}$ should not be excessively large to prevent significant rotation of the motor rotor during the detection period. Additionally, they should provide sufficient magnetizing and demagnetizing effects to accurately distinguish the N/S poles of the rotor. But a larger injection voltage and injection time may cause motor rotation. Therefore, repeated experiments and adjustments are necessary to select a more appropriate voltage magnitude and injection time. $T_{o f f}$ only ensure that the response current returns to zero during that period.

The feedback virtual $d * - axis$ current $i_{d} *$ can be obtained by utilizing the Clark and Park transformation to convert the value of $i_{a}$ , $i_{b}$ , $i_{c}$ , as shown in equation (10):

i_{d} * = \frac{2}{3} [i_{a} \cos θ_{c} + i_{b} \cos (θ_{c} - \frac{2}{3} π) + i_{c} \cos (θ_{c} + \frac{2}{3} π)]

(10)

where

i_{a}

i_{b}

i_{c}

are the sampled three-phase currents.

A block diagram illustrating the voltage pulse injection method is shown in Figure 7. The injection voltage sequence is illustrated in Figure 8. $θ_{R 1}$ ∼ $θ_{R 5}$ represent the angles corresponding to the maximum current of each round, and V1∼V24 are the injection voltage vectors in Figure 8(a)∼(e).

The specific process of the voltage pulse injection method is shown in Figure 9. $U_{m}$ is the value of injection voltage vectors, $i d_{m a x} *$ represents the largest value of $i_{d} *$ in each round, $θ_{V}$ is the angular position of the injection voltage vector, $i d_{n} *$ is the peak magnitude of the current response corresponding to each voltage vector. In the first round, the voltage vectors V1∼V12 are injected from 0° to 360° at intervals of $Δ θ_{V}$ , which is 30°. To reduce the rotation of the motor during the detection process, the injection sequence in Figure 8(a) is adopted instead of sequentially injecting the voltage vectors from 0° to 360°. By comparing the feedback peak current values, the angle $θ_{R 1}$ corresponding to the maximum current value of the first round is selected as the output angle for the first round. In the second round, the voltage vectors corresponding to the three angles $θ_{R 1}$ , $θ_{R 1} + 0.5 Δ θ_{V}$ , $θ_{R 1} - 0.5 Δ θ_{V}$ are selected for the second round of injection. The angle $θ_{R 2}$ corresponding to the maximum feedback peak current is chosen as the output angle for the second round. Thus, through steps (a)∼(e) in Figure 8, the final initial angle detection result $θ_{R 5}$ is obtained. By continuously narrowing the search range, the voltage pulse injection method can theoretically achieve detection results with extremely high precision.

Drawbacks of the traditional voltage pulse injection method

Wrong judgment of pole direction

The judgment of the magnetic poles of this method is based on the magnetizing and demagnetizing effects of the stator magnetic field. The d-axis inductance is smaller when the stator magnetic field is aligned with the rotor's N pole, compared to when it is in the opposite direction. Therefore, the feedback current corresponding to $Δ θ$ = 0° is greater than that at $Δ θ$ = 180°. Thus, the N/S poles of the rotor can be distinguished by the magnitude of the feedback current.

However, to prevent significant rotation during the detection, the magnitude of the injected voltage and duration are not set too large, which may result in insufficient magnetizing and demagnetizing effects. Consequently, the current produced by the voltage aligned with the d-axis may be smaller than that produced by the opposite voltage, causing a judgment error of 180° in the detection results.

Easily affected by the accuracy of hardware current detection

From equation (5), differentiating $i_{d} *$ with respect to $Δ θ$ yields equation (11):

\overset{\approx}{\dot{i_{d} *}} \frac{U}{R_{s}} E \sin (2 Δ θ)

(11)

It can be observed from Equation (11) that as $Δ θ$ gradually approaches 0, the derivative of $i_{d} *$ also approaches 0. Therefore, when $Δ θ$ is near 0, the difference in $i_{d} *$ between distinct $Δ θ$ values gradually diminishes. The voltage pulse injection method relies on comparing the values of $i_{d} *$ to detect the initial position. At this point, the existence of sampling errors may lead to incorrect judgments, thereby reducing detection accuracy. Thus, the accuracy of current sampling significantly impacts the detection accuracy of this method.

Proposal of an improved method

To improve the detection accuracy of the voltage pulse injection method, the following solutions are introduced.

Add the magnetic pole identification (MPI) program. After completing step (a) in Figure 8, the detection result of the round $θ_{R 1}$ can be obtained. To verify whether this result is correct and locate near the actual initial angle rather than near the opposite direction of the actual angle. Two voltage vectors at angles of $θ_{R 1}$ and $θ_{R 1} + π$ are injected, and comparing their values of $i_{d} *$ . To increase the difference of current at the N/S pole, enhance the magnitude of the injection voltage and duration during MPI. Since the angle of the injection voltage is very close to $θ_{e}$ , so $Δ θ$ is relatively low. According to equation (12), increasing the value of the injection voltage will not produce excessive electromagnetic torque that would cause the rotor to rotate.

T_{e} = 1.5 P_{n} ψ_{f} i_{d} * \sin (Δ θ)

(12)

It can be inferred from Section II that a larger injection voltage and duration can increase the degree of magnetic flux saturation, thereby reducing the value of $L_{d}$ , allowing for a larger $i_{d} *$ , and enhancing the differences in $i_{d} *$ generated by injection voltages at various angles. Therefore, in the processes (b), (c), (d), and (e) following the MPI, the magnitude and duration can be moderately increased to reduce the impact of sampling errors and the disturbances on the sampling results, thereby improving position detection accuracy.

Using the obtained current and angle information for curve fitting, the initial electrical angle can be calculated. As described by the expression for $i_{d} *$ in equation (5), the relationship between the angle and the current forms a quadratic curve when $Δ θ$ approaches 0, as shown in Figure 10. To maximize the utilization of the acquired information, this quadratic curve is fitted to detect the initial electrical angle. After completing the process (e) (where the angle is subdivided to 1.875 degrees), utilizing the angle information corresponding to the voltage vectors V24, V25, and V26, along with $i d_{24} *$ , $i d_{25} *$ , $i d_{26} *$ to perform Lagrange curve fitting. After fitting the curve $i_{d} *$ regarding the angle, the angle corresponding to the maximum value is selected as the final initial position detection result.

Lagrange second-order interpolation polynomial:

P (x) = \sum_{k = 0}^{2} l_{k} (x) y (k)

(13)

l_{k} (x) = \prod_{j = 0, j \neq k}^{2} \frac{x - x_{j}}{x_{k} - x_{j}}

(14)

Quadratic equation expression:

y = a x^{2} + b x + c

(15)

Substituting the angle information $θ_{24}$ , $θ_{25}$ , $θ_{26}$ and the corresponding $i d_{24} *$ , $i d_{25} *$ , $i d_{26} *$ into equations (13) and (14), and combining with equation (15), a quadratic polynomial describing the relationship between the angle and the current value can be obtained.

A = \frac{i d_{24} *}{θ_{24}^{2} - θ_{26} θ_{24} - θ_{25} θ_{24} + θ_{25} θ_{26}}

(16)

B = \frac{i d_{25} *}{θ_{25}^{2} - θ_{26} θ_{25} - θ_{25} θ_{24} + θ_{24} θ_{26}}

(17)

C = \frac{i d_{26} *}{θ_{26}^{2} - θ_{25} θ_{26} - θ_{24} θ_{26} + θ_{25} θ_{24}}

(18)

i d^{*} = (A + B + C) θ^{2} - (A θ_{25} + A θ_{26} + B θ_{24} + B θ_{26} + C θ_{24} + C θ_{25}) θ + (A θ_{25} θ_{26} + B θ_{24} θ_{26} + C θ_{24} θ_{25})

(19)

Based on the extreme value formula of the quadratic function, the angle corresponding to the maximum value of $i_{d} *$ is:

θ = \frac{(A θ_{25} + A θ_{26} + B θ_{24} + B θ_{26} + C θ_{24} + C θ_{25})}{2 (A + B + C)}

(20)

This value is taken as the final initial position detection result. The specific process of the improved method is shown in Figure 11.

The curve fitting algorithm is analyzed by using Simulink. The motor model parameters used in the simulation are as follows: number of pole pairs is 5, phase resistance is 0.2 Ω, d- and q-axis inductances are 0.0004 H, and the rotor flux linkage is 0.0086 V·s. In the simulation, $U_{m} = 5 V$ , $T_{o n} = 0.0002 s$ , and the results are shown in Figure 12. As shown in Figure 12, the fitted curve agrees well with the sampled data points, which verifies the effectiveness of the curve fitting algorithm.

Experimental results and analysis

The motor used in the experiment is SPMSM. The parameters of the selected motor for the experiment are shown in Table 1. The experimental platform employed is presented in Figure 13. The motor platform is primarily composed of a control circuit and a power circuit. The control circuit uses TI's DSP C2000 series, which integrates modules including ADC, PWM, and encoder modules. The power circuit employs TI's DRV8305 chip. In this experiment, the current sampling period of the motor control system is 0.00005 s.

Table 1.

Motor parameter.

Parameter	Value
Number Of Poles(pairs)	5
Line Resistance(Ω)	0.4
Line Inductance(mH)	0.7
Rotor Moment Of Inertia( $k g . m^{2} \times 10^{- 4}$ )	0.32
Rated power(w)	200
Rated Speed(rpm)	3000
Rated Voltage(V)	36

First, the appropriate voltage injection duration and amplitude are determined through experimental testing. The final selected parameters are $U_{m} = 9 V$ , $T_{o n} = 0.00055 s$ . The computational complexity is O(n), the memory requirements is 3 KB, and the algorithm execution time is 1.314 s.

The initial position detection experiment was conducted after the motor rotor was driven to an electrical angle of 200° using the magnetic pole orientation method. In the experiment, the motor rotor exhibits slight vibrations.

Figure 14 shows the peak feedback current curve using the traditional voltage pulse injection method. Among them, the feedback current value generated by V4 (210° voltage vector) is 10.26, theoretically expected to be the largest among the current values generated by the first round of injected voltage vectors V1∼V12. But as seen in Figure 14, the feedback current value generated by V3 (30° voltage vector) is 11.203, which is the largest among the current values generated by V1∼V12. This leads to the incorrect judgment that 30° is the angle closest to the initial position in the first round. This is due to insufficient magnetizing and demagnetizing effects, which fail to distinguish the N/S poles of the rotor, resulting in magnetic pole misidentification in the first round. After injecting the voltage vector V24, the final detection result was 46.88°, indicating magnetic pole misidentification.

Figure 15 shows the peak feedback current curve using the traditional voltage pulse injection method with MPI. From Figure 15(a), it can be observed that the feedback current value generated by V3 (30° voltage vector) is still greater than that generated by V4 (210° voltage vector). However, by incorporating MPI, the improved method uses a larger injection voltage magnitude and a longer injection duration to enhance the magnetizing and demagnetizing effects, resulting in a significant difference in feedback current at the N/S poles. As shown in Figure 15(a), it can be seen that the feedback current value generated by V13 (210° voltage vector) is much greater than that generated by V14 (30° voltage vector), thereby correcting the erroneous identification result and eliminating the magnetic pole misidentification. Due to the influence of slight vibrations during the detection process, the electrical angle of the rotor has changed from 200° to 201.98°. Using traditional methods with MPI, after injecting the voltage vector V26, the detection result was 210°.

Figure 16 shows the peak feedback current curve using the improved voltage pulse injection method. It can be observed that the differences in feedback currents between V15, V16, and V17 in Figure 16(b) are larger than those in Figure 15(b), reducing the impact of sampling errors on the results. Due to the influence of slight vibrations during the detection process, the electrical angle of the rotor has changed from 200° to 202.34°. Using the improved method, after injecting the voltage vector V26, the angle detection result was 198.8°. This result is better than the result of the traditional voltage pulse injection method with MPI.

In Figure 17, the red line represents the current curve versus $Δ θ$ obtained through curve fitting, based on the values of $i d_{24} *$ , $i d_{25} *$ , $i d_{26} *$ and angles corresponding to voltage vectors V24, V25, and V26 in Figure 16(b). The angle value at the extremum point is calculated using the extremum formula (20), yielding a final detection result of 199.3°. This result is closer to the changed electrical angle 202.34°, compared to the detection result of 198.8° without curve fitting.

The detection result from the traditional voltage pulse injection method is applied to the current closed-loop control system, and the corresponding position feedback curve is shown in Figure 18. Similarly, the detection result from the improved method is implemented in the same closed-loop control system, yielding the position feedback curve presented in Figure 19. As shown in Figure 18, the motor reverses during startup due to the magnetic pole misidentification in the detection results. In contrast, Figure 19 confirms that the motor starts successfully and operates normally with the improved method.

Select 10 angles between 0° and 360° as the angles to be tested, conduct 10 experiments for each angle, and record the detection errors. The detection error of the improved method compared to the traditional method with MPI is shown in Figure 20. Compared to the traditional method that only incorporates MPI, the proposed improved method reduces the angle detection error by 32% on average.

Finally, the connecting shaft between the controlled motor and the load motor was removed to evaluate the performance of the improved voltage pulse injection method under no-load conditions. The initial angle was set to 280°, and the magnitude and duration of the injected voltage vector were kept the same as those designed in the previous section. The current feedback results are shown in Figure 21, and the rotor electrical angular displacement during the detection process is shown in Figure 22. Due to the influence of slight vibrations during the detection process, the electrical angle of the rotor has changed from 280° to 284.32°, and the detected initial rotor position was 288.4°. Under motor no-load operation, the moment of inertia is small. Therefore, it is required to decrease both the amplitude and action time of the injected voltage to restrict over-rotation of the rotor.

Conclusion

In practical applications, the traditional voltage pulse injection method may result in magnetic pole misidentification and limited accuracy. To solve these problems, an improved voltage pulse injection method is proposed. By MPI, the correct direction of the motor rotor is confirmed, eliminating magnetic pole misidentification. The influence of sampling error is reduced by enhancing the magnetizing effect through increasing the magnitude and duration of the injection voltage after MPI. The application of Lagrange interpolation maximizes the use of the acquired information and effectively improves detection accuracy. Finally, the effectiveness of the improved method was validated through experiments at multiple electrical angles. The experimental results demonstrate a successful elimination of magnetic pole misidentification, and the average detection accuracy is improved by 32%.

Footnotes

ORCID iDs

Keping Liu

Piao Fan

Ethical considerations

This article does not contain any studies with human or animal participants.

Consent to participate

Not applicable

Consent for publication

Not applicable

Author contributions

Keping Liu: Conceptualization, Methodology, Software, Investigation, Formal Analysis; Yan Xu: Data Curation, Writing-Original Draft; Yan Li: Visualization, Writing-Review&Editing; Piao Fan: Resources, Supervision; Jianlei Fan: Software, Validation.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Department of Science and Technology in Jilin Province under Grant (number.20250203169SF).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

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