﻿ 车用电机控制器功能安全及主动短路分析
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (9): 1298-1305.  DOI: 10.11908/j.issn.0253-374x.2018.09.018 0

### 引用本文

WU Zhihong, LU Ke, ZHU Yuan. Analysis of Active-Short-Circuit of Permanent Magnet Synchronous Motor in Electric Vehicles[J]. Journal of Tongji University (Natural Science), 2018, 46(9): 1298-1305. DOI: 10.11908/j.issn.0253-374x.2018.09.018

### 文章历史

1. 同济大学 汽车学院，上海 201804;
2. 同济大学 中德学院，上海 200092

Analysis of Active-Short-Circuit of Permanent Magnet Synchronous Motor in Electric Vehicles
WU Zhihong 1, LU Ke 1, ZHU Yuan 2
1. School of Automotive Engineering, Tongji University, Shanghai 201804, China;
2. Sino-German College, Tongji University, Shanghai 200092, China
Abstract: According to the mathematical model of permanent magnet synchronous motor, this paper uses the analytical method to obtain the current and torque equation of the permanent magnet synchronous motor under the active short-circuit operation in different working conditions, and reveals the relationship between the motor output and motor parameters. The results are verified by simulations and experiments. The significance of this paper is that the advantages and disadvantages of using active short-circuiting as a safety shut-off path is analyzed theoretically, and the current calculation formula for permanent magnet synchronous motor under active short-circuit condition is proposed, which provides a basis for the motor controller to select a suitable power device.
Key words: permanent magnetics synchronous machine(PMSM)    functional safety    active-short-circuit    shut-off path

1 电动汽车驱动电机功能安全和安全状态

 图 1 电动汽车动力总成框图 Fig.1 Block diagram of powertrain of electric vehicle

 图 2 三相电压型逆变器的拓扑结构 Fig.2 Topology structure of three phase voltage inverter

2 永磁同步电机模型及主动短路

 $\frac{{{\rm{d}}\mathit{\boldsymbol{i}}}}{{{\rm{d}}t}} = \mathit{\boldsymbol{Ai}} + \mathit{\boldsymbol{Bu}}$ (1)
 $\mathit{\boldsymbol{i}} = \left[ \begin{array}{l} {i_d}\\ {i_q} \end{array} \right]$ (2)
 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} { - \frac{{{R_{\rm{s}}}}}{{{L_d}}}}&{\frac{{\omega {L_q}}}{{{L_d}}}}\\ { - \frac{{\omega {L_d}}}{{{L_q}}}}&{ - \frac{{{R_{\rm{s}}}}}{{{L_q}}}} \end{array}} \right]$ (3)
 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{L_d}}}}&0\\ 0&{\frac{1}{{{L_q}}}} \end{array}} \right]$ (4)
 $\mathit{\boldsymbol{u}} = \left[ {\begin{array}{*{20}{c}} {{u_d}}\\ {{u_q} - \omega {\mathit{\Psi }_{\rm{f}}}} \end{array}} \right]$ (5)

dq坐标轴系下的永磁同步电机的转矩方程为

 ${T_{\rm{e}}} = 1.5{p_{\rm{n}}}[{\mathit{\Psi }_{\rm{f}}}{i_q} + ({L_d} - {L_q}){i_d}{i_q}]$ (6)

 图 3 永磁同步电机FOC控制算法框图 Fig.3 Block diagram of FOC control algorithm for permanent magnet synchronous motor
3 永磁同步电机主动短路电流及转矩响应分析

 $\mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }} = \mathit{\boldsymbol{i}} - \left[ {\begin{array}{*{20}{c}} { - \frac{{{\omega ^2}{L_q}{\mathit{\Psi }_{\rm{f}}}}}{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}}}\\ { - \frac{{\omega {R_{\rm{s}}}{\mathit{\Psi }_{\rm{f}}}}}{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}}} \end{array}} \right]$ (7)

 $\frac{{{\rm{d}}\mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}}}{{{\rm{d}}t}} = \mathit{\boldsymbol{A\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}$ (8)

 $\mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}\left( t \right) = {{\rm{e}}^{At}}\mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}\left( 0 \right)$ (9)

 $\mathit{\boldsymbol{A}} = {\mathit{\boldsymbol{T}}^{ - 1}}\mathit{\boldsymbol{JT}}$ (10)

 $\mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}\left( t \right) = \mathit{\boldsymbol{T}}{{\rm{e}}^{\mathit{\boldsymbol{J}}t}}{\mathit{\boldsymbol{T}}^{ - 1}}\mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}\left( 0 \right)$ (11)

 $\begin{array}{*{20}{c}} \mathit{\boldsymbol{J}} ={\left( { - \frac{{( - \sqrt { - ({L_d}{R_{\rm{s}}} - {L_q}{R_{\rm{s}}} + 2\omega {L_d}{L_q})({L_q}{R_{\rm{s}}} - {L_d}{R_{\rm{s}}} + 2\omega {L_d}{L_q})} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2{L_d}{L_q}}}} \right.}\\ 0\\ 0\\ {\left. { - \frac{{(\sqrt { - ({L_d}{R_{\rm{s}}} - {L_q}{R_{\rm{s}}} + 2\omega {L_d}{L_q})({L_q}{R_{\rm{s}}} - {L_d}{R_{\rm{s}}} + 2\omega {L_d}{L_q})} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2{L_d}{L_q}}}} \right)} \end{array}$ (12)
 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{T}} = \left( { - \frac{{{R_{\rm{s}}}}}{{\omega {L_d}}} + \frac{{( - \sqrt { - ({L_d}{R_{\rm{s}}} - {L_q}{R_{\rm{s}}} + 2\omega {L_d}{L_q})({L_q}{R_{\rm{s}}} - {L_d}{R_{\rm{s}}} + 2\omega {L_d}{L_q})} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2\omega {L_d}}}} \right.}\\ 1\\ {\left. {\frac{{(\sqrt { - ({L_d}{R_{\rm{s}}} - {L_q}{R_{\rm{s}}} + 2\omega {L_d}{L_q})({L_q}{R_{\rm{s}}} - {L_d}{R_{\rm{s}}} + 2\omega {L_d}{L_q})} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2\omega {L_d}}} - \frac{{{R_{\rm{s}}}}}{{\omega {L_d}}}} \right)}\\ 1 \end{array}$ (13)

 $\begin{array}{l} ( - ({L_d}{R_{\rm{s}}} - {L_q}{R_{\rm{s}}} + 2\omega {L_d}{L_q}) \cdot \\ ({L_q}{R_{\rm{s}}} - {L_d}{R_{\rm{s}}} + 2\omega {L_d}{L_q}){)^{\frac{1}{2}}} \approx \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;2\omega {L_d}{L_q}{\rm{j}} \end{array}$ (14)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{J}} \approx \left( { - \frac{{( - 2\omega {L_d}{L_q}{\rm{j}} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2{L_d}{L_q}}}} \right.}\\ 0\\ 0\\ {\left. { - \frac{{(2\omega {L_d}{L_q}{\rm{j}} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2{L_d}{L_q}}}} \right)} \end{array}$ (15)
 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{T}} \approx \left( {\frac{{( - 2\omega {L_d}{L_q}{\rm{j}} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2\omega L_d^2}} - \frac{{{R_{\rm{s}}}}}{{\omega {L_d}}}} \right.}\\ 1\\ {\left. {\frac{{(2\omega {L_d}{L_q}{\rm{j}} + {L_d}{R_{\rm{s}}} + {L_q}{R_{\rm{s}}})}}{{2\omega L_d^2}} - \frac{{{R_{\rm{s}}}}}{{\omega {L_d}}}} \right)}\\ 1 \end{array}$ (16)

 $\begin{array}{l} \mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}\left( t \right) \approx \left[ {{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t}}{{2{L_d}}}}}{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t}}{{2{L_q}}}}}(2\omega L_q^2{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\sin \left( {\omega t} \right) + } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;{L_d}{R_{\rm{s}}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\sin \left( {\omega t} \right) - {L_q}{R_{\rm{s}}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\sin \left( {\omega t} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;2\omega {L_d}{L_q}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\cos \left( {\omega t} \right))/2\omega {L_d}{L_q}\\ \;\;\;\;\;\;\;\;\;\;\; - {{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t}}{{2{L_d}}}}}{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t}}{{2{L_q}}}}}(2\omega {L_d}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\sin \left( {\omega t} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;{L_d}{R_{\rm{s}}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\sin \left( {\omega t} \right) - {L_q}{R_{\rm{s}}}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\sin \left( {\omega t} \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\left. {2\omega {L_d}{L_q}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\cos \left( {\omega t} \right)/2\omega {L_d}{L_q}} \right] \end{array}$ (17)

 $\begin{array}{l} \mathit{\boldsymbol{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }}\left( t \right) \approx \\ \left( {\begin{array}{*{20}{c}} {\frac{{{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t\left( {{L_d} + {L_q}} \right)}}{{2{L_d}{L_q}}}}}({L_d}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\cos \left( {\omega t} \right) + {L_q}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\sin \left( {\omega t} \right))}}{{{L_d}}}}\\ {\frac{{{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t\left( {{L_d} + {L_q}} \right)}}{{2{L_d}{L_q}}}}}({L_q}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\cos \left( {\omega t} \right) - {L_d}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\sin \left( {\omega t} \right))}}{{{L_q}}}} \end{array}} \right) \end{array}$ (18)

 $\begin{array}{l} \mathit{\boldsymbol{i}}\left( t \right) \approx \\ \left( {\begin{array}{*{20}{c}} {\frac{{{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t\left( {{L_d} + {L_q}} \right)}}{{2{L_d}{L_q}}}}}({L_d}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\cos \left( {\omega t} \right) + {L_q}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\sin \left( {\omega t} \right))}}{{{L_d}}}}\\ {\frac{{{{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t\left( {{L_d} + {L_q}} \right)}}{{2{L_d}{L_q}}}}}({L_q}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{q0}}\cos \left( {\omega t} \right) - {L_d}{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} }_{d0}}\sin \left( {\omega t} \right))}}{{{L_q}}}} \end{array}} \right) + \\ \left( {\begin{array}{*{20}{c}} { - \frac{{{\omega ^2}{L_q}{\mathit{\Psi }_{\rm{f}}}}}{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}}}\\ { - \frac{{\omega {R_{\rm{s}}}{\mathit{\Psi }_{\rm{f}}}}}{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}}} \end{array}} \right) \end{array}$ (19)

 $\mathit{\boldsymbol{I}}\left( t \right) \approx \left( \begin{array}{l} - \frac{{{\omega ^2}{L_q}{\mathit{\Psi }_{\rm{f}}}}}{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}} \pm {{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t\left( {{L_d} + {L_q}} \right)}}{{2{L_d}{L_q}}}}} \cdot \\ \sqrt {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} _{d0}^2 + \frac{{L_q^2\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} _{q0}^2}}{{L_d^2}}} \\ - \frac{{\omega {R_{\rm{s}}}{\mathit{\Psi }_{\rm{f}}}}}{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}} \pm {{\rm{e}}^{ - \frac{{{R_{\rm{s}}}t\left( {{L_d} + {L_q}} \right)}}{{2{L_d}{L_q}}}}} \cdot \\ \sqrt {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} _{q0}^2 + \frac{{L_d^2\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over i} _{d0}^2}}{{L_q^2}}} \end{array} \right)$ (20)

 ${T_{\rm{e}}} = - 1.5{p_{\rm{n}}}\left( {\frac{{\omega {R_{\rm{s}}}\mathit{\Psi }_{\rm{f}}^2}}{{{L_d}{L_q}{\omega ^2}}} + R_{\rm{s}}^2 - \frac{{{\omega ^3}{L_q}{R_{\rm{s}}}\mathit{\Psi }_{\rm{f}}^2({L_d} - {L_q})}}{{{{({L_d}{L_q}{\omega ^2} + R_{\rm{s}}^2)}^2}}}} \right)$ (21)

 ${T_{\rm{e}}} = - 1.5{p_{\rm{n}}}\left( {\frac{{{R_{\rm{s}}}\mathit{\Psi }_{\rm{f}}^2}}{{{L_d}{L_q}\omega }} - \frac{{{R_{\rm{s}}}\mathit{\Psi }_{\rm{f}}^2({L_d} - {L_q})}}{{L_d^2{L_q}\omega }}} \right)$ (22)

 ${T_{\rm{e}}} = - 1.5{p_{\rm{n}}}\left( {\frac{{\omega \mathit{\Psi }_{\rm{f}}^2}}{{{R_{\rm{s}}}}} - \frac{{{\omega ^3}{L_q}\mathit{\Psi }_{\rm{f}}^2({L_d} - {L_q})}}{{R_{\rm{s}}^3}}} \right)$ (23)

4 仿真和实验验证

 图 4 4 000 r·min-1时主动短路估算结果 Fig.4 Estimated result of active short circuit at 4 000 r·min-1
 图 5 4 000 r·min-1时主动短路仿真结果 Fig.5 Simulation result of active short circuit at 4 000 r·min-1
 图 6 4 000 r·min-1时主动短路实验结果 Fig.6 Experiment result of active short circuit at 4 000 r·min-1
 图 7 3 000 r·min-1时主动短路估算结果 Fig.7 Estimated result of active short circuit at 3 000 r·min-1
 图 8 3 000 r·min-1时主动短路仿真结果 Fig.8 Simulation result of active short circuit at 3 000 r·min-1
 图 9 3 000 r·min-1时主动短路实验结果 Fig.9 Experiment result of active short circuit at 3 000 r·min-1
 图 10 2 000 r·min-1时主动短路估算结果 Fig.10 Estimated result of active short circuit at 2 000 r·min-1
 图 11 2 000 r·min-1时主动短路仿真结果 Fig.11 Simulation result of active short circuit at 2 000 r·min-1
 图 12 2 000 r·min-1时主动短路实验结果 Fig.12 Experiment result of active short circuit at 2 000 r·min-1

5 结语

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