变换坐标的艺术—PMSM扩展反电势公式推导
本文推导Zhiqian Chen扩展反电势数学模型。
PMSM在 d q dq dq轴的电压方程为 [ u d u q ] = R [ i d i q ] [ p L d 0 0 p L q ] [ i d i q ] ω e [ ? ψ q ψ d ] \left[ \begin{array}{c} u_d\\ u_q\\ \end{array} \right] =R\left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] \left[ \begin{matrix} pL_d& 0\\ 0& pL_q\\ \end{matrix} \right] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] \omega _e\left[ \begin{array}{c} -\psi _q\\ \psi _d\\ \end{array} \right] [uduq]=R[idiq]+[pLd00pLq][idiq]+ωe[−ψqψd]
构造次对角线对称矩阵，可得： [ u d u q ] = [ R + p L d 0 0 R + p L q ] [ i d i q ] + ω e [ − L q i q L d i d + ψ f ] \left[ \begin{array}{c} u_d\\ u_q\\ \end{array} \right] =\left[ \begin{matrix} R+pL_d& 0\\ 0& R+pL_q\\ \end{matrix} \right] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] +\omega _e\left[ \begin{array}{c} -L_qi_q\\ L_di_d+\psi _f\\ \end{array} \right] [uduq]=[R+pLd00R+pLq][idiq]+ωe[−LqiqLdid+ψf] 进一步整理，可得： [ u d u q ] = [ R + p L d − ω e L q 0 R + p L q ] [ i d i q ] + ω e [ 0 L d i d + ψ f ] \left[ \begin{array}{c} u_d\\ u_q\\ \end{array} \right] =\left[ \begin{matrix} R+pL_d& -\omega _eL_q\\ 0& R+pL_q\\ \end{matrix} \right] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] +\omega _e\left[ \begin{array}{c} 0\\ L_di_d+\psi _f\\ \end{array} \right] [uduq]=[R+pLd0−ωeLqR+pLq][idiq]+ωe[0Ldid+ψf] 再次整理，可得： [ u d u q ] = [ R + p L d − ω e L q ω e L q R + p L d ] [ i d i q ] + [ 0 ( L d − L q ) ( ω e i d − p i q ) + ω e ψ f ] \left[ \begin{array}{c} u_d\\ u_q\\ \end{array} \right] =\left[ \begin{matrix} R+pL_d& -\omega _eL_q\\ \omega _eL_q& R+pL_d\\ \end{matrix} \right] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] +\left[ \begin{array}{c} 0\\ \left( L_d-L_q \right) \left( \omega _ei_d-pi_q \right) +\omega _e\psi _f\\ \end{array} \right] [uduq]=[R+pLdωeLq−ωeLqR+pLd][idiq]+[0(Ld−Lq)(ωeid−piq)+ωeψf] 为了便于后续推导，拆分矩阵可得： [ u d u q ] = [ R − ω e L q ω e L q R ] [ i d i q ] + [ p L d 0 0 p L d ] [ i d i q ] + [ 0 ( L d − L q ) ( ω e i d − p i q ) + ω e ψ f ] \left[ \begin{array}{c} u_d\\ u_q\\ \end{array} \right] =\left[ \begin{matrix} R& -\omega _eL_q\\ \omega _eL_q& R\\ \end{matrix} \right] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] +\left[ \begin{matrix} pL_d& 0\\ 0& pL_d\\ \end{matrix} \right] \left[ \begin{array}{c} i_d\\ i_q\\ \end{array} \right] +\left[ \begin{array}{c} 0\\ \left( L_d-L_q \right) \left( \omega _ei_d-pi_q \right) +\omega _e\psi _f\\ \end{array} \right] [uduq]=[RωeLq−ωeL