﻿ 永磁同步电机的转子初始位置检测

# 永磁同步电机的转子初始位置检测Rotor Initial Position Detection of Permanent Magnet Synchronous Motor

Abstract: Aiming at poor reliability and small applicable range of the traditional rotor initial position detection method for permanent magnet synchronous motor, a rotor position detection method using high frequency signal injection is improved. The high frequency signal is injected in the three-phase permanent magnet synchronous motor. The salient effect based on the current con-taining rotor position information is obtained and filtered, where a band-pass filter based on proportional resonance controller is designed, which simplifies the structure of the filter and is easy to realize digitally. Then, the position information is extracted by a heterodyne method and the position is accurately estimated by the rotor position observer. The simulation results show that the presented method can accurately estimate the rotor position and the average error of position detection is within 0.1˚.

1. 引言

2. 高频激励下三相永磁同步电机的数学模型

$\begin{array}{c}{u}_{d}=R{i}_{d}+\frac{\text{d}}{\text{d}t}{\psi }_{d}-{\omega }_{e}{\psi }_{q}\\ {u}_{q}=R{i}_{q}+\frac{\text{d}}{\text{d}t}{\psi }_{q}+{\omega }_{e}{\psi }_{d}\end{array}$ (1)

$\left\{\begin{array}{c}{\psi }_{d}={L}_{d}{i}_{d}+{\psi }_{f}\\ {\psi }_{q}={L}_{q}{i}_{q\begin{array}{c}\end{array}}\begin{array}{cc}& \end{array}\end{array}$ (2)

$\left\{\begin{array}{c}{u}_{d}=R{i}_{d}+{L}_{d}\frac{\text{d}}{\text{d}t}{i}_{d}-{\omega }_{e}{L}_{q}{i}_{q}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}\\ {u}_{q}=R{i}_{q}+{L}_{q}\frac{\text{d}}{\text{d}t}{i}_{q}+{\omega }_{e}\left({L}_{d}{i}_{d}+{\psi }_{f}\right)\end{array}$ (3)

$\left[\begin{array}{c}{u}_{ds}\\ {u}_{qs}\end{array}\right]=R\left[\begin{array}{c}{i}_{ds}\\ {i}_{qs}\end{array}\right]+\frac{\text{d}}{\text{d}t}\left[\begin{array}{c}{\psi }_{ds}\\ {\psi }_{qs}\end{array}\right]$ (4)

$\left[\begin{array}{c}{\psi }_{ds}\\ {\psi }_{qs}\end{array}\right]=\left[\begin{array}{cc}L+\Delta L\mathrm{cos}2{\theta }_{r}& \Delta L\mathrm{sin}2{\theta }_{r}\\ \Delta L\mathrm{sin}2{\theta }_{r}& L-\Delta L\mathrm{cos}2{\theta }_{r}\end{array}\right]\left[\begin{array}{c}{i}_{ds}\\ {i}_{qs}\end{array}\right]+{\psi }_{f}\left[\begin{array}{c}\mathrm{cos}{\theta }_{r}\\ \mathrm{sin}{\theta }_{r}\end{array}\right]$ (5)

$\Delta L=\left({L}_{d}-{L}_{q}\right)/2$ —半差电感。

${L}_{dqs}=\left[\begin{array}{cc}L+\Delta L\mathrm{cos}2{\theta }_{r}& \Delta L\mathrm{sin}2{\theta }_{r}\\ \Delta L\mathrm{sin}2{\theta }_{r}& L-\Delta L\mathrm{cos}2{\theta }_{r}\end{array}\right]$ (6)

$\left\{\begin{array}{c}{u}_{dhs}\approx {L}_{d}\frac{\text{d}{i}_{dhs}}{\text{d}t}\\ {u}_{qhs}\approx {L}_{q}\frac{\text{d}{i}_{qhs}}{\text{d}t}\end{array}$ (7)

$\left\{\begin{array}{c}{T}_{e}=\frac{2}{3}{n}_{p}\left({\psi }_{d}{i}_{d}-{\psi }_{q}{i}_{q}\right)\\ \frac{Jp{\omega }_{e}}{{n}_{p}}={T}_{e}-{T}_{L}-{R}_{\Omega }\frac{{\omega }_{e}}{{n}_{p}}\end{array}$ (8)

3. 基于高频信号注入法的转子位置估计

3.1. 高频电压信号的选择

3.2. 基于比例谐振控制器的带通滤波器设计

3.2.1. 比例谐振控制原理

${G}_{PR}{\left(s\right)}_{1}=\frac{1}{2}\left[{G}_{PI}\left(s+j{\omega }_{0}\right)+{G}_{PI}\left(s-j{\omega }_{0}\right)\right]={K}_{p}+\frac{2{K}_{i}s}{s+{\omega }_{0}^{2}}$ (9)

Figure 1. Quasi-PR controller block diagram

${G}_{PR}\left(s\right)={K}_{p}+\frac{2{K}_{i}{\omega }_{c}s}{{s}^{2}+2{\omega }_{c}s+{\omega }_{0}^{2}}$ (10)

3.2.2. 比例谐振带通滤波器设计方法

$s=\frac{{\omega }_{0}\left(z-1\right)}{\mathrm{tan}\left({\omega }_{0}{T}_{s}/2\right)\left(z+1\right)}$ (11)

${G}_{PR}\left(z\right)=\frac{{b}_{0}+{b}_{1}z+{b}_{2}{z}^{2}}{{a}_{0}+{a}_{1}z+{a}_{2}{z}^{2}}$ (12)

Figure 2. The result of changing ${K}_{p}$

Figure 3. The result of changing ${K}_{i}$

Figure 4. The result of changing ${\omega }_{c}$

$\begin{array}{l}{b}_{0}=\frac{-d{k}_{i}{\omega }_{c}}{{d}^{2}+d{\omega }_{c}+{\omega }_{0}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{2}=\frac{d{k}_{i}{\omega }_{c}}{{d}^{2}+d{\omega }_{c}+{\omega }_{0}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{1}=0\\ {a}_{0}=\frac{{d}^{2}-d{\omega }_{c}+{\omega }_{0}^{2}}{{d}^{2}+d{\omega }_{c}+{\omega }_{0}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}=\frac{2\left({\omega }_{c}^{2}-{d}^{2}\right)}{{d}^{2}+d{\omega }_{c}+{\omega }_{0}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{2}=1\end{array}$

$y\left(k\right)={b}_{0}x\left(k-2\right)+{b}_{2}x\left(k\right)-{a}_{1}y\left(k-1\right)-{a}_{0}y\left(k-2\right)$ (13)

3.3. 旋转高频电压激励下永磁同步电机的电流响应

${u}_{\alpha \beta hs}=\left[\begin{array}{c}{u}_{\alpha hs}\\ {u}_{\beta hs}\end{array}\right]=\left[\begin{array}{c}{V}_{in}\mathrm{cos}{\omega }_{in}t\\ {V}_{in}\mathrm{sin}{\omega }_{in}t\end{array}\right]={V}_{in}{\text{e}}^{j{\omega }_{in}t}$ (14)

${u}_{dqhs}={u}_{\alpha \beta hs}{e}^{-j{\theta }_{r}}={V}_{in}{\text{e}}^{j{\omega }_{in}t}$ (15)

$\begin{array}{l}{i}_{dqhs}=\frac{{V}_{in}}{{L}_{d}}\int \mathrm{cos}\left({\omega }_{in}t-{\theta }_{r}\right)\text{d}t+j\frac{{V}_{in}}{{L}_{q}}\\ \begin{array}{ccc}& =& \frac{{V}_{in}}{{\omega }_{in}{L}_{d}{L}_{q}}\end{array}\left[\frac{{L}_{d}+{L}_{q}}{2}{\text{e}}^{j\left({\omega }_{in}t-{\theta }_{r}-\frac{\text{π}}{\text{2}}\right)}+\frac{{L}_{d}-{L}_{q}}{2}{\text{e}}^{j\left(-{\omega }_{in}t+{\theta }_{r}+\frac{\text{π}}{\text{2}}\right)}\right]\end{array}$ (16)

${i}_{\alpha \beta hs}={i}_{dq}{e}^{j{\theta }_{r}}={I}_{cp}{\text{e}}^{j\left({\omega }_{in}t-\frac{\text{π}}{2}\right)}+{I}_{cn}{\text{e}}^{j\left(-{\omega }_{in}t+2{\theta }_{r}+\frac{\text{π}}{2}\right)}$ (17)

${I}_{cn}$ —负相序高频电流分量的幅值， ${I}_{cn}=\frac{{V}_{in}}{{\omega }_{in}{L}_{d}{L}_{q}}\frac{{L}_{d}-{L}_{q}}{2}$

Figure 5. Corresponding current response system diagram in the motor

3.4. 转子位置估计的实现方法

${i}_{dqhs_n}={I}_{cn}{\text{e}}^{j\left(-{\omega }_{in}t+2{\theta }_{r}+\frac{\text{π}}{2}\right)}$ (18)

$\begin{array}{l}\epsilon ={i}_{qhs_n}\mathrm{cos}\left(2{\stackrel{^}{\theta }}_{r}-{\omega }_{in}t+\frac{\text{π}}{\text{2}}\right)-{i}_{dhs_n}\mathrm{sin}\left(2{\stackrel{^}{\theta }}_{r}-{\omega }_{in}t+\frac{\text{π}}{\text{2}}\right)\\ \begin{array}{c}\end{array}={I}_{cn}\mathrm{sin}\left(2{\theta }_{r}-2{\stackrel{^}{\theta }}_{r}\right)\end{array}$ (19)

4. 仿真建模与结果分析

Figure 6. Rotor position detection schematic

Figure 7. Three-phase PMSM initial position detection simulation of rotating high-frequency voltage signal injection

Table 1. Simulation PMSM parameter table

Figure 8. Simulation result of PMSM at 30˚ initial position

Figure 9. Position error at a PMSM initial angle of 30˚

Figure 10. PMSM rotor position offset during inspection

5. 结论

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