CN104660140A - Permanent magnet synchronous motor initial position detection method based on high-frequency current signal injection - Google Patents

Abstract

The invention discloses a method for detecting the initial position of a permanent magnet synchronous motor based on high-frequency current signal injection. This method is based on the initial initial position estimation using the pulse vibration high-frequency current injection method, and then extracts the positive direction information of the d-axis contained in the sub-harmonic of the frequency of the double-injected signal in the d-axis voltage response to determine the positive direction of the d-axis. Judgment, this method only needs to inject high-frequency current signals in the entire initial position estimation process, and the d-axis positive direction judgment process does not need other forms of additional signal injection such as square wave pulses, which shortens the estimation time and simplifies the estimation process.

Description

Initial position detection method of permanent magnet synchronous motor based on high frequency current signal injection

technical field

The invention relates to the field of motor control, in particular to a method for detecting the initial position of a permanent magnet synchronous motor based on high-frequency current signal injection.

Background technique

For the method of detecting the initial position of the surface-mounted permanent magnet synchronous motor rotor, since the motor rotor is stationary, the initial position of the rotor cannot be estimated by detecting the back EMF of the motor. High-frequency signal injection method.

Liu Ying, Zhou Bo, Li Shuai, et al. Rotor initial position detection of surface-mounted permanent magnet synchronous motor[J]. Proceedings of the Chinese Society for Electrical Engineering, 2011, 31(18): 48-54. Estimation of rotor synchronous rotation coordinates The d-axis of the system injects a high-frequency sinusoidal voltage signal, detects the q-axis high-frequency current response and establishes a position estimation closed loop to obtain the initial estimated value of the rotor position. Then inject positive and negative voltage pulses in the direction of the estimated d-axis, and use the difference in the equivalent time constant of the direct axis under the action of positive and negative currents to judge the positive direction of the d-axis. This method needs to inject two types of signals during the entire process of initial position estimation. , the implementation is more complicated. The process of judging the positive direction of the magnetic pole needs to inject positive and negative voltage pulses and then compare the time it takes for the current response to decay to 0, which also takes a certain amount of time. Liu Ying, Zhou Bo, Zhao Chengliang, et al. Low-speed position sensorless control of SPMSM based on pulse-vibration high-frequency current injection[J]. Chinese Journal of Electrotechnical Society, 2012, 7(27):139-145. The first use of pulse-vibration high-frequency current injection The SPMSM rotor position estimation method is realized, but there is no mention of how to judge the positive direction of the d-axis. Wang Jianmin, Zhao Xiaomin, Guo Zhen. Analysis and Improvement of Pulsating Current Injection Based Sensorless Control of Permanent Magnet Synchronous Motor[C]. 17th International Conference on Electrical Machines and Systems (ICEMS), 2014. Also implemented by pulse vibration high frequency current injection method The detection of the initial position of the rotor, and some solutions to reduce the estimation error are given, but it does not mention how to judge the positive direction of the d-axis.

According to the existing literature review, there is no article dedicated to the judgment of the positive direction of the d-axis in the pulse vibration high-frequency current injection method.

Contents of the invention

The technical problem to be solved by the present invention is aimed at the defects of the background technology. On the basis of the existing initial position detection method, a d A method for judging the positive direction of the axis, which significantly simplifies the initial position detection process.

The present invention adopts the following technical solutions for solving the problems of the technologies described above:

A method for detecting the initial position of a permanent magnet synchronous motor based on high-frequency current signal injection, comprising the following steps:

Step 1), constructing a rotor position closed-loop adjustment system based on the pulse vibration high-frequency current injection method, and obtaining the initial estimated value of the rotor position;

Step 2), extracting the second harmonic component of the d-axis voltage response, performing corresponding signal modulation on it, judging the positive direction of the d-axis, and obtaining the compensation value after judging the positive direction of the d-axis;

In step 3), the initial estimated value of the rotor position is added to the compensation value after judging the positive direction of the d-axis to obtain the final estimated value of the initial position.

As the first process of the method for detecting the initial position of a permanent magnet synchronous motor based on high-frequency current signal injection in the present invention, the steps for obtaining the initial estimated value of the rotor position are as follows:

Step 11), the q-axis current of the estimated rotor synchronous rotation coordinate system is given as 0, and the d-axis current is given as a pulse vibration high-frequency sinusoidal signal I _mh sin(ω _h t), wherein, I _mh is on the d-axis The amplitude of the injected high-frequency current, ω _h is the angular frequency of the injected high-frequency current on the d-axis, and t represents the current moment;

Step 12), using the proportional resonant controller PR to control the estimated d-axis current and q-axis current to make it consistent with the given;

Step 13), compare the output voltage of the proportional resonance controller and Perform Park inverse transformation to obtain the voltages u _α and u _β in the two-phase static α-β coordinate system, and then use the space vector pulse width modulation strategy to obtain the six-way switching signals of the three-phase inverter to drive the surface-mounted permanent magnet synchronous motor ;

Step 14), detect any two-phase current in the three-phase winding A/B/C of the motor, first perform Clarke transformation to obtain the currents i _α and i _β in the two-phase static α-β coordinate system, and then obtain the estimated rotor through Park transformation d-axis current in synchronous rotating coordinate system and q-axis current Feed it back to the input of the proportional resonant controller;

Step 15), the q-axis voltage response of the rotor synchronous rotating coordinate system will be estimated The AC component with a frequency of ω _h is selected through a band-pass filter, which is the first harmonic component of the q-axis voltage response Then multiplied with the cosine signal cos(ω _h t) for modulation to obtain a DC component and an AC component with a frequency of 2ω _h. Finally, the AC component is filtered out by a low-pass filter, and the DC component is extracted to obtain an estimated position deviation signal f _c ( Δθ);

Step 16), build a position deviation closed loop, use the estimated position deviation signal f _c (Δθ) as the input of the proportional-integral regulator, and estimate the rotor angular velocity is the output of the proportional-integral regulator, for the estimated rotor angular velocity Integrate to get the estimated rotor position;

Step 17), repeat steps A.1) to A.6), until the estimated rotor position converges to a constant value, which is the initial estimated value of the initial rotor position

As the second process of the method for detecting the initial position of a permanent magnet synchronous motor based on high-frequency current signal injection in the present invention, the detailed steps of judging the positive direction of the d-axis and obtaining the compensation value after judging the positive direction of the d-axis are as follows:

Step 21), detecting and estimating the d-axis voltage response of the rotor synchronously rotating coordinate system Pass it through a band-pass filter to select the second harmonic component of the d-axis voltage response

Step 22), the second harmonic component Multiply with the sinusoidal signal sin(2ω _h t) for modulation to obtain a DC component and an AC component with a frequency of 4ω _h , where ω _h is the angular frequency of the high-frequency current signal injected on the d-axis, and t represents the current moment;

Step 23), filter the modulated signal through a low-pass filter to remove the AC component, and extract the DC component, which is the d-axis positive direction judgment information g _NS .

Furthermore, the positive direction of the d-axis is judged according to the sign of g _NS . If g _NS is less than 0, the positive direction of the d-axis is in the same direction as the N pole of the magnetic pole, and the compensation value after the judgment of the positive direction of the d-axis is 0; if g _NS is greater than 0, the positive direction of the d-axis is opposite to the N pole of the magnetic pole, and the compensation value after the positive direction of the d-axis is judged is π.

The present invention uses the traditional pulse-vibration high-frequency current injection method to realize the initial initial position estimation, extracts and estimates the information in the subharmonic of the frequency of the injected signal twice in the estimated d-axis high-frequency voltage response, and judges the positive direction of the d-axis. In the whole process of initial position estimation, only high-frequency current signals need to be injected, and in the process of judging the positive direction of the d-axis, there is no need for other forms of signal injection such as square waves, which reduces the complexity of the operation, shortens the estimation time, and simplifies the initial position estimation process.

Compared with the prior art, the present invention adopts the above technical scheme and has the following technical effects:

(1) In the process of judging the positive direction of the d-axis, the injected current is always a sinusoidal high-frequency signal, and there is no need to inject other forms of signals such as positive and negative pulses, which simplifies the estimation process;

(2) Use a band-pass filter to obtain the second harmonic component of the estimated d-axis high-frequency voltage response, and then extract the component containing the positive direction information of the d-axis from the second harmonic through a certain modulation, and judge according to the sign of this component In the positive direction of the d-axis, this process can be completed within one cycle of injecting the current signal, which shortens the time required for the process of judging the positive direction of the d-axis.

Description of drawings

Figure 1 is a schematic block diagram of the initial position estimation process of the surface-mounted permanent magnet synchronous motor rotor;

Fig. 2 is a schematic diagram of the relative relationship between the two-phase stationary coordinate system, the actual two-phase synchronous rotating coordinate system and the estimated two-phase synchronous rotating coordinate system;

Fig. 3 is the functional block diagram of the signal extraction and modulation process of initial initial position estimation;

Fig. 4 is the functional block diagram of the signal extraction and modulation process of d-axis positive direction judgment;

Figure 5(a) corresponds to the simulation waveform of the initial rotor position estimation process when the actual rotor position is 1 rad;

Figure 5(b) corresponds to the simulation waveform of the positive direction judgment information g _NS of the d-axis when the actual rotor position is 1rad;

Figure 5(c) corresponds to the simulation waveform of the initial rotor position estimation process when the actual rotor position is 3rad;

Figure 5(d) corresponds to the simulation waveform of the judgment information g _NS in the positive direction of the d-axis when the actual rotor position is 3rad.

Figure 6(a) corresponds to the actual rotor position of 0.5 rad, the initial rotor position estimation process and the experimental waveform of the d-axis positive direction judgment information g _NS ;

Figure 6(b) corresponds to the experimental waveform of the initial rotor position estimation process and the d-axis positive direction judgment information g _NS when the actual rotor position is 4rad.

Detailed ways

Below in conjunction with accompanying drawing, technical scheme of the present invention is described in further detail:

As shown in Figure 1, the present invention provides a method for detecting the initial position of a surface-mounted permanent magnet synchronous motor, which specifically includes the following steps:

Step 1), establish the coordinate system relationship diagram, as shown in Figure 2, dq is the actual synchronous rotating coordinate system, In order to estimate the rotor synchronous rotating coordinate system, α-β is the actual two-phase stationary coordinate system, and define the estimated position error Among them, θ is the actual rotor initial position, is the final initial position estimate, The initial value of is 0.

Step 2), as shown in Figure 3, the q-axis current of the estimated rotor synchronous rotation coordinate system is given as 0, and the d-axis current is given as a pulse vibration high-frequency sinusoidal signal I _mh sin(ω _h t), where, I _mh is the amplitude of the high-frequency current injected on the d-axis, ω _h is the angular frequency of the high-frequency current injected on the d-axis, and t represents the current moment.

Step 3), using a proportional resonant controller (PR) to control the estimated d-axis current and q-axis current to make it consistent with the given.

Step 4), the output voltage of the PR regulator and Perform Park inverse transformation to obtain the voltages u _α and u _β in the two-phase static α-β coordinate system, and then use the space vector pulse width modulation strategy to obtain the six-way switching signals of the three-phase inverter to drive the surface-mounted permanent magnet synchronous motor SPMSM.

Step 5), detect any two-phase current in the three-phase winding A/B/C of the motor, as shown in Figure 1, this embodiment chooses to detect the A-phase and B-phase current, and first performs Clarke transformation to obtain the two-phase static α The current i _α and i _β in the -β coordinate system, and then through the Park transformation to obtain the estimated d-axis current in the rotor synchronous rotation coordinate system and q-axis current Feed it back to the PR regulator; similarly, detecting currents of phase B and phase C, and phase A and phase C can also achieve the same effect of the initial position detection method, which will not be described here.

Step 6), the q-axis voltage response of the rotor synchronously rotating coordinate system will be estimated The AC component with the frequency ω _h selected by the band-pass filter is the first harmonic component of the q-axis voltage response Then multiplied with the cosine signal cos(ω _h t) for modulation to obtain a DC component and an AC component with a frequency of 2ω _h. Finally, the AC component is filtered out by a low-pass filter, and the DC component is extracted to obtain an estimated position deviation signal f _c ( Δθ).

Step 7), build a position deviation closed loop, use the estimated position deviation signal f _c (Δθ) as the input of the PI regulator, and estimate the rotor angular velocity is the output of the PI regulator, the estimated rotor angular velocity Integrate to obtain the estimated rotor position, repeat steps 2-7 until the estimated rotor position converges to a constant value, which is the initial estimated value of the initial rotor position

Step 8), as shown in Figure 4, first adopts the band-pass filter to estimate the d-axis high-frequency voltage response from Select the harmonic component of twice the frequency of the injected signal from Multiply it with a sinusoidal signal sin(2ω _h t) for modulation, and then use a low-pass filter to filter out high-frequency components and extract DC components to obtain the d-axis square judgment information g _NS . Judge the positive direction of the d-axis according to the sign of g _NS . If g _NS is less than 0, the positive direction of the d-axis is in the same direction as the N pole of the magnetic pole, and the compensation value after judging the positive direction of the d-axis is 0, that is If g _NS is greater than 0, the positive direction of the d-axis is opposite to the N pole of the magnetic pole, and the compensation value after judging the positive direction of the d-axis is π, that is

The theoretical analysis of the method for judging the positive direction of the d-axis is as follows:

For surface-mounted permanent magnet synchronous motors, when the d-axis current acts as a magnetizer, the magnetic field is saturated and the inductance decreases; when the d-axis current acts as a demagnetization, the magnetic field desaturates and the inductance increases. In order to simplify the analysis, assuming that the d-axis current is positive, the average value of the inductance is and remain unchanged; when the d-axis current is negative, the average value of the inductance is According to the saturation characteristics of the d-axis inductance, it is obvious that Right now Liu Ying, Zhou Bo, Zhao Chengliang, et al. Low-speed position sensorless control of SPMSM based on pulse vibration high-frequency current injection[J]. After the current, the d-axis voltage response of the permanent magnet synchronous motor in the estimated synchronous rotating coordinate system is:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; Z</span>}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&Delta;\&theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

In the formula, is the average impedance, is the semi-difference impedance, Z _d ＝ _rs + jω _h L _d is the d-axis high-frequency impedance, Z _q ＝ _rs + jω _h L _q is the q-axis high-frequency impedance, rs _is the stator resistance, L _d is the d-axis Inductance, L _q is the q-axis inductance.

After the initial initial position estimation is completed, the position estimation error Δθ is 0 or π, which can be substituted into formula (1) to get:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\sqrt{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arctan</span>}\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

Assuming that Δθ is 0, considering the change of d-axis inductance, formula (2) can be further sorted out:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\sqrt{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\text{<span class="notranslate"> t+arctan</span>}\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; ,</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; )</span>\\ \sqrt{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arctan</span>}\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; ,</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Carry out Fourier series expansion on formula (3), and ignore the third and above harmonics to get:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

According to the Fourier series correlation theory, the correlation coefficient in formula (4) is calculated: a ₀ =0, b ₁ = _rs I _mh , a ₂ =0, Substituting these coefficients into equation (4) gives:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

It can be seen from formula (5) that since the d-axis inductance changes with the current, it is estimated that there is a second harmonic component in the d-axis voltage, and Δθ corresponding to formula (5) is 0. If Δθ is π, similarly, formula (3) should be changed to:

${{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\sqrt{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\text{<span class="notranslate"> t+arctan</span>}\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; ,</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; )</span>\\ \sqrt{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arctan</span>}\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}}{{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; ,</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; k\&pi;</span>}}{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Obviously, formula (3) and formula (6) satisfy:

${{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&pi;</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

According to formula (5) and formula (7), when Δθ is π, formula (5) should be rewritten as:

${{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

Comparing formula (5) and formula (8), when the position estimation error Δθ is 0, the coefficient in front of the second harmonic term in formula (5) is negative, when the position estimation error Δθ is π, formula (8) The coefficient in front of the second harmonic term in is positive, defining the function s(Δθ):

$\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&Delta;\&theta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; \&Delta;\&theta;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; \&Delta;\&theta;</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; \&pi;</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

Then formula (5) and formula (8) can be uniformly expressed as:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}}_{\mathrm{<span\; class="notranslate">\; d\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}}{\mathrm{<span\; class="notranslate">\; 3</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; -</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{<span\; class="notranslate">\; +</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; m\; h</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; \&omega;</span>}}_{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

It can be known from formula (10) that the magnitude sign of the second harmonic component of the estimated d-axis high-frequency voltage response is related to the judgment result of the positive direction of the d-axis. If the positive direction of the d-axis is in the same direction as the magnetic pole N, that is, Δθ=0, the sign of the amplitude is less than 0; if the positive direction of the d-axis is opposite to the magnetic pole N, that is, Δθ=π, the sign of the amplitude is greater than 0.

In order to verify the correctness of the proposed method, the simulation is carried out using matlab software and the experiment is carried out based on the hardware platform, as follows.

As shown in Fig. 5(a) and Fig. 5(b), the simulation waveform corresponding to the initial position estimation process and the d-axis positive direction judgment information g _NS when the actual rotor initial position is 1 rad. g _NS is less than 0, indicating that the positive direction of the d-axis is in the same direction as the magnetic pole N, and there is no need to estimate the position for the first time Perform angle compensation, θ _c = 0; as shown in Figure 5(c) and Figure 5(d), corresponding to the simulation waveform of the initial position estimation process and d-axis positive direction judgment information g _NS when the actual rotor initial position is 3rad. g _NS is greater than 0, indicating that the positive direction of the d-axis is opposite to the N pole of the magnetic pole, and it is necessary to estimate the position for the first time Compensate for π radians, θ _c = π, and the final initial position estimate is

As shown in Fig. 6(a), it corresponds to the experimental waveform of the initial position estimation process and the positive direction judgment information g _NS of the d-axis when the actual rotor initial position is 0.5 rad. g _NS is less than 0, indicating that the positive direction of the d-axis is in the same direction as the magnetic pole N, and there is no need to estimate the position for the first time Perform angle compensation, θ _c =0; as shown in Figure 6(b), it corresponds to the simulation waveform of the initial position estimation process and the d-axis positive direction judgment information g _NS when the actual rotor initial position is 4rad. g _NS is greater than 0, indicating that the positive direction of the d-axis is opposite to the N pole of the magnetic pole, and it is necessary to estimate the position for the first time Compensate for π radians, θ _c = π, and the final initial position estimate is

Claims (4)

1., based on the permagnetic synchronous motor initial position detection method that high-frequency current signal injects, it is characterized in that, comprise following steps:

Step 1), build rotor-position closed-loop regulating system based on pulsating high frequency current injection, obtain the first estimated value of rotor-position;

Step 2), extract the second harmonic component of d shaft voltage response, corresponding signal madulation is carried out to it, judges d axle positive direction, obtain the offset after the judgement of d axle positive direction;

Step 3), first for rotor-position estimated value is added the offset after the judgement of d axle positive direction, obtain final initial position estimation value.

2. the permagnetic synchronous motor initial position detection method injected based on high-frequency current signal according to claim 1, is characterized in that, the step obtaining the first estimated value of rotor-position is as follows:

Step 11), the q shaft current of estimated rotor synchronous rotating frame is given as 0, d shaft current and is given as a pulsating high frequency sinusoidal signal I _mhsin (ω _ht), wherein, I _mhfor injecting the amplitude of high-frequency current at d axle, ω _hfor injecting the angular frequency of high-frequency current at d axle, t represents current time;

Step 12), adoption rate resonant controller PR controls the d shaft current estimated and q shaft current, makes it consistent with given;

Step 13), the voltage that comparative example resonant controller PR exports with carry out Park inverse transformation, obtain the voltage u under the static alpha-beta coordinate system of two-phase _αand u _β, then adopt space vector pulse width modulation strategy to obtain six path switching signals of three-phase inverter, the mounted permagnetic synchronous motor of table-drive;

Step 14), detect any biphase current in motor three-phase windings A/B/C, first carry out Clarke conversion obtain the static alpha-beta coordinate system of two-phase under current i _αand i _β, then the d shaft current under Park conversion obtains estimated rotor synchronous rotating frame with q shaft current fed back to the input of ratio resonant controller;

Step 15), the q shaft voltage of estimated rotor synchronous rotating frame is responded selecting frequency through band pass filter is ω _halternating current component, be q shaft voltage response first harmonic component again with cosine signal cos (ω _ht) be multiplied and modulate, obtain DC component and frequency is 2 ω _halternating current component, eventually pass low pass filter filtering alternating current component, extract DC component, obtain estimated position deviation signal f _c(Δ θ);

Step 16), build position deviation closed loop, by estimated position deviation signal f _c(Δ θ) as the input of proportional and integral controller, estimated rotor angular speed for the output of proportional and integral controller, to estimated rotor angular speed integration obtains the rotor-position estimated;

Step 17), repeat step 11) to step 16), until the rotor-position estimated converges to a steady state value, be the first estimated value of initial position of rotor

3. the permagnetic synchronous motor initial position detection method injected based on high-frequency current signal according to claim 1, is characterized in that, described judgement d axle positive direction and obtain d axle positive direction judge after the detailed step of offset as follows:

Step 21), detect the d shaft voltage response of estimated rotor synchronous rotating frame it is first selected the second harmonic component of d shaft voltage response through band pass filter

Step 22), by second harmonic component with sinusoidal signal sin (2 ω _ht) be multiplied and modulate, obtain DC component and frequency is 4 ω _halternating current component, wherein ω _hfor injecting the angular frequency of high-frequency current signal at d axle, t represents current time;

Step 23), by the signal after modulation through low pass filter filtering alternating current component, extract DC component, this DC component is d axle positive direction and judges information g _nS.

4. the permagnetic synchronous motor initial position detection method injected based on high-frequency current signal according to claim 3, is characterized in that, according to g _nSsymbol decision d axle positive direction, if g _nSbe less than 0, then the N pole of d axle positive direction and magnetic pole in the same way, and the offset after d axle positive direction judges is 0; If g _nSbe greater than 0, then the N of d axle positive direction and magnetic pole is extremely reverse, and the offset after d axle positive direction judges is π.