坐标变换的艺术—估计轴系的扩展反电势公式推导

Shigeo Morimoto将扩展反电势拓展至估计轴系，本文将对其进行公式演绎。

公式推导过程

坐标变换图示

注：论文中的角度误差使用θe\theta_eθe表示，本文则使用θ~\tilde{\theta}θ~表示

两相静止轴系，电机电压方程为
[uαuβ]=[R00R][iαiβ]+ddt{[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe][iαiβ]}+ωeψf[−sin⁡θecos⁡θe]\left[ \begin{array}{c} u_{\alpha}\\ u_{\beta}\\ \end{array} \right] =\left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\alpha}\\ i_{\beta}\\ \end{array} \right] +\frac{\mathrm{d}}{\mathrm{dt}}\left\{ \left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\alpha}\\ i_{\beta}\\ \end{array} \right] \right\} +\omega _e\psi _f\left[ \begin{array}{c} -\sin \theta _e\\ \cos \theta _e\\ \end{array} \right] [uαuβ]=[R00R][iαiβ]+dtd{[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe][iαiβ]}+ωeψf[−sinθecosθe]
式中：
θ~=θe−θ^，L0=Ld+Lq2，L1=Lq−Ld2\tilde{\theta}=\theta _e-\hat{\theta}\text{，}L_0=\frac{L_d+L_q}{2}\text{，} L_1=\frac{L_q-L_d}{2} θ~=θe−θ^，L0=2Ld+Lq，L1=2Lq−Ld
坐标变换矩阵满足
[fαfβ]=[cos⁡θe−sin⁡θesin⁡θecos⁡θe][fdfq]=Tdq/αβ[fdfq][fαfβ]=[cos⁡θ^−sin⁡θ^sin⁡θ^cos⁡θ^][fγfδ]=Tγδ/αβ[fγfδ]\left[ \begin{array}{c} f_{\alpha}\\ f_{\beta}\\ \end{array} \right] =\left[ \begin{matrix} \cos \theta _e& -\sin \theta _e\\ \sin \theta _e& \cos \theta _e\\ \end{matrix} \right] \left[ \begin{array}{c} f_d\\ f_q\\ \end{array} \right] =T_{dq/\alpha \beta}\left[ \begin{array}{c} f_d\\ f_q\\ \end{array} \right] \\ \left[ \begin{array}{c} f_{\alpha}\\ f_{\beta}\\ \end{array} \right] =\left[ \begin{matrix} \cos \hat{\theta}& -\sin \hat{\theta}\\ \sin \hat{\theta}& \cos \hat{\theta}\\ \end{matrix} \right] \left[ \begin{array}{c} f_{\gamma}\\ f_{\delta}\\ \end{array} \right] =T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} f_{\gamma}\\ f_{\delta}\\ \end{array} \right] [fαfβ]=[cosθesinθe−sinθecosθe][fdfq]=Tdq/αβ[fdfq][fαfβ]=[cosθ^sinθ^−sinθ^cosθ^][fγfδ]=Tγδ/αβ[fγfδ]
同时，坐标变换矩阵是单位正交矩阵，可得
T−1=T−TT^{-1}=T^{-T} T−1=T−T
因此，电压方程可重写为
Tγδ/αβ[uγuδ]=[R00R]Tγδ/αβ[iγiδ]+ddt{[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe]Tγδ/αβ[iγiδ]}+ωeψf[−sin⁡θecos⁡θe]T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =\left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\frac{\mathrm{d}}{\mathrm{dt}}\left\{ \left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \right\} +\omega _e\psi _f\left[ \begin{array}{c} -\sin \theta _e\\ \cos \theta _e\\ \end{array} \right] Tγδ/αβ[uγuδ]=[R00R]Tγδ/αβ[iγiδ]+dtd{[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe]Tγδ/αβ[iγiδ]}+ωeψf[−sinθecosθe]
整理可得：
[uγuδ]=Tγδ/αβ−1[R00R]Tγδ/αβ[iγiδ]+Tγδ/αβ−1ddt{[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe]Tγδ/αβ[iγiδ]}+Tγδ/αβ−1ωeψf[−sin⁡θecos⁡θe]\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =T_{\gamma \delta /\alpha \beta}^{-1}\left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +T_{\gamma \delta /\alpha \beta}^{-1}\frac{\mathrm{d}}{\mathrm{dt}}\left\{ \left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \right\} +T_{\gamma \delta /\alpha \beta}^{-1}\omega _e\psi _f\left[ \begin{array}{c} -\sin \theta _e\\ \cos \theta _e\\ \end{array} \right] [uγuδ]=Tγδ/αβ−1[R00R]Tγδ/αβ[iγiδ]+Tγδ/αβ−1dtd{[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe]Tγδ/αβ[iγiδ]}+Tγδ/αβ−1ωeψf[−sinθecosθe]
对等式右边的式子逐一进行整理，第一项
Tγδ/αβ−1[R00R]Tγδ/αβ[iγiδ]⇒[cos⁡θ^sin⁡θ^−sin⁡θ^cos⁡θ^][R00R][cos⁡θ^−sin⁡θ^sin⁡θ^cos⁡θ^][iγiδ]⇒[R00R][iγiδ]（记为A[iγiδ]）\begin{aligned} &T_{\gamma \delta /\alpha \beta}^{-1}\left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \\ \Rightarrow& \left[ \begin{matrix} \cos \hat{\theta}& \sin \hat{\theta}\\ -\sin \hat{\theta}& \cos \hat{\theta}\\ \end{matrix} \right] \left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] \left[ \begin{matrix} \cos \hat{\theta}& -\sin \hat{\theta}\\ \sin \hat{\theta}& \cos \hat{\theta}\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \\ \Rightarrow& \left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \text{（记为}A\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \text{）} \end{aligned} ⇒⇒Tγδ/αβ−1[R00R]Tγδ/αβ[iγiδ][cosθ^−sinθ^sinθ^cosθ^][R00R][cosθ^sinθ^−sinθ^cosθ^][iγiδ][R00R][iγiδ]（记为A[iγiδ]）
第二项
Tγδ/αβ−1ddt{[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe]Tγδ/αβ[iγiδ]}⇒Tγδ/αβ−1ddt{[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe]}Tγδ/αβ[iγiδ]+Tγδ/αβ−1[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe]ddt{Tγδ/αβ}[iγiδ]+Tγδ/αβ−1[L0−L1cos⁡2θe−L1sin⁡2θe−L1sin⁡2θeL0+L1cos⁡2θe]Tγδ/αβddt{[iγiδ]}⇒[2L1sin⁡(2θe−2θ^)−2L1cos⁡(2θe−2θ^)−2L1cos⁡(2θe−2θ^)−2L1sin⁡(2θe−2θ^)]ωe[iγiδ]+[−L1sin⁡(2θe−2θ^)−L0+L1cos⁡(2θe−2θ^)L0+L1cos⁡(2θe−2θ^)L1sin⁡(2θe−2θ^)]ω^e[iγiδ]+[L0−L1cos⁡(2θe−2θ^)−L1sin⁡(2θe−2θ^)−L1sin⁡(2θe−2θ^)L0+L1cos⁡(2θe−2θ^)]ddt[iγiδ]\begin{aligned} &T_{\gamma \delta /\alpha \beta}^{-1}\frac{\mathrm{d}}{\mathrm{dt}}\left\{ \left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \right\} \\ \Rightarrow& T_{\gamma \delta /\alpha \beta}^{-1}\frac{\mathrm{d}}{\mathrm{dt}}\left\{ \left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] \right\} T_{\gamma \delta /\alpha \beta}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +T_{\gamma \delta /\alpha \beta}^{-1}\left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] \frac{\mathrm{d}}{\mathrm{dt}}\left\{ T_{\gamma \delta /\alpha \beta} \right\} \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +T_{\gamma \delta /\alpha \beta}^{-1}\left[ \begin{matrix} L_0-L_1\cos 2\theta _e& -L_1\sin 2\theta _e\\ -L_1\sin 2\theta _e& L_0+L_1\cos 2\theta _e\\ \end{matrix} \right] T_{\gamma \delta /\alpha \beta}\frac{\mathrm{d}}{\mathrm{dt}}\left\{ \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \right\} \\ \Rightarrow& \left[ \begin{matrix} 2L_1\sin \left( 2\theta _e-2\hat{\theta} \right)& -2L_1\cos \left( 2\theta _e-2\hat{\theta} \right)\\ -2L_1\cos \left( 2\theta _e-2\hat{\theta} \right)& -2L_1\sin \left( 2\theta _e-2\hat{\theta} \right)\\ \end{matrix} \right] \omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} -L_1\sin \left( 2\theta _e-2\hat{\theta} \right)& -L_0+L_1\cos \left( 2\theta _e-2\hat{\theta} \right)\\ L_0+L_1\cos \left( 2\theta _e-2\hat{\theta} \right)& L_1\sin \left( 2\theta _e-2\hat{\theta} \right)\\ \end{matrix} \right]\hat{\omega}_e \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} L_0-L_1\cos \left( 2\theta _e-2\hat{\theta} \right)& -L_1\sin \left( 2\theta _e-2\hat{\theta} \right)\\ -L_1\sin \left( 2\theta _e-2\hat{\theta} \right)& L_0+L_1\cos \left( 2\theta _e-2\hat{\theta} \right)\\ \end{matrix} \right] \frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \\ \end{aligned} ⇒⇒Tγδ/αβ−1dtd{[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe]Tγδ/αβ[iγiδ]}Tγδ/αβ−1dtd{[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe]}Tγδ/αβ[iγiδ]+Tγδ/αβ−1[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe]dtd{Tγδ/αβ}[iγiδ]+Tγδ/αβ−1[L0−L1cos2θe−L1sin2θe−L1sin2θeL0+L1cos2θe]Tγδ/αβdtd{[iγiδ]}⎣⎡2L1sin(2θe−2θ^)−2L1cos(2θe−2θ^)−2L1cos(2θe−2θ^)−2L1sin(2θe−2θ^)⎦⎤ωe[iγiδ]+⎣⎡−L1sin(2θe−2θ^)L0+L1cos(2θe−2θ^)−L0+L1cos(2θe−2θ^)L1sin(2θe−2θ^)⎦⎤ω^e[iγiδ]+⎣⎡L0−L1cos(2θe−2θ^)−L1sin(2θe−2θ^)−L1sin(2θe−2θ^)L0+L1cos(2θe−2θ^)⎦⎤dtd[iγiδ]
为了辅助推导，将第二项的推导结果简记为
⇒Bωe[iγiδ]+Cω^[iγiδ]+Dddt[iγiδ]\Rightarrow B\omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +C\hat{\omega}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +D\frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] ⇒Bωe[iγiδ]+Cω^[iγiδ]+Ddtd[iγiδ]
第三项
Tγδ/αβ−1ωeψf[−sin⁡θecos⁡θe]⇒[cos⁡θ^sin⁡θ^−sin⁡θ^cos⁡θ^]ωeψf[−sin⁡θecos⁡θe]⇒[−sin⁡(θe−θ^)cos⁡(θe−θ^)]ωeψf（记为Eωeψf）\begin{aligned} &T_{\gamma \delta /\alpha \beta}^{-1}\omega _e\psi _f\left[ \begin{array}{c} -\sin \theta _e\\ \cos \theta _e\\ \end{array} \right] \\ \Rightarrow& \left[ \begin{matrix} \cos \hat{\theta}& \sin \hat{\theta}\\ -\sin \hat{\theta}& \cos \hat{\theta}\\ \end{matrix} \right] \omega _e\psi _f\left[ \begin{array}{c} -\sin \theta _e\\ \cos \theta _e\\ \end{array} \right] \\ \Rightarrow& \left[ \begin{array}{c} -\sin \left( \theta _e-\hat{\theta} \right)\\ \cos \left( \theta _e-\hat{\theta} \right)\\ \end{array} \right] \omega _e\psi _f\text{（记为}E\omega _e\psi _f\text{）} \end{aligned} ⇒⇒Tγδ/αβ−1ωeψf[−sinθecosθe][cosθ^−sinθ^sinθ^cosθ^]ωeψf[−sinθecosθe]⎣⎡−sin(θe−θ^)cos(θe−θ^)⎦⎤ωeψf（记为Eωeψf）
进一步化简A、B、C、D、EA、B、C、D、EA、B、C、D、E，可得：
A=[R00R]B=[2L1sin⁡(2θe−2θ^)−2L1cos⁡(2θe−2θ^)−2L1cos⁡(2θe−2θ^)−2L1sin⁡(2θe−2θ^)]=[2(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)(1−2sin⁡2θ~)−(Lq−Ld)(1−2sin⁡2θ~)−2(Lq−Ld)sin⁡θ~cos⁡θ~]C=[−L1sin⁡(2θe−2θ^)−L0+L1cos⁡(2θe−2θ^)L0+L1cos⁡(2θe−2θ^)L1sin⁡(2θe−2θ^)]=[−(Lq−Ld)sin⁡θ~cos⁡θ~−Ld−(Lq−Ld)sin⁡2θ~Lq−(Lq−Ld)sin⁡2θ~(Lq−Ld)sin⁡θ~cos⁡θ~]D=[L0−L1cos⁡(2θe−2θ^)−L1sin⁡(2θe−2θ^)−L1sin⁡(2θe−2θ^)L0+L1cos⁡(2θe−2θ^)]=[Ld+(Lq−Ld)sin⁡2θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡θ~cos⁡θ~Lq−(Lq−Ld)sin⁡2θ~]E=[−sin⁡(θe−θ^)cos⁡(θe−θ^)]=[−sin⁡θ~cos⁡θ~]\begin{aligned} A&=\left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] \\ B&=\left[ \begin{matrix} 2L_1\sin \left( 2\theta _e-2\hat{\theta} \right)& -2L_1\cos \left( 2\theta _e-2\hat{\theta} \right)\\ -2L_1\cos \left( 2\theta _e-2\hat{\theta} \right)& -2L_1\sin \left( 2\theta _e-2\hat{\theta} \right)\\ \end{matrix} \right] =\left[ \begin{matrix} 2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -\left( L_q-L_d \right) \left( 1-2\sin ^2\tilde{\theta} \right)\\ -\left( L_q-L_d \right) \left( 1-2\sin ^2\tilde{\theta} \right)& -2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \\ C&=\left[ \begin{matrix} -L_1\sin \left( 2\theta _e-2\hat{\theta} \right)& -L_0+L_1\cos \left( 2\theta _e-2\hat{\theta} \right)\\ L_0+L_1\cos \left( 2\theta _e-2\hat{\theta} \right)& L_1\sin \left( 2\theta _e-2\hat{\theta} \right)\\ \end{matrix} \right] =\left[ \begin{matrix} -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -L_d-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}& \left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \\ D&=\left[ \begin{matrix} L_0-L_1\cos \left( 2\theta _e-2\hat{\theta} \right)& -L_1\sin \left( 2\theta _e-2\hat{\theta} \right)\\ -L_1\sin \left( 2\theta _e-2\hat{\theta} \right)& L_0+L_1\cos \left( 2\theta _e-2\hat{\theta} \right)\\ \end{matrix} \right] =\left[ \begin{matrix} L_d+\left( L_q-L_d \right) \sin ^2\tilde{\theta}& -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ \end{matrix} \right] \\ E&=\left[ \begin{array}{c} -\sin \left( \theta _e-\hat{\theta} \right)\\ \cos \left( \theta _e-\hat{\theta} \right)\\ \end{array} \right] =\left[ \begin{array}{c} -\sin \tilde{\theta}\\ \cos \tilde{\theta}\\ \end{array} \right] \end{aligned} ABCDE=[R00R]=⎣⎡2L1sin(2θe−2θ^)−2L1cos(2θe−2θ^)−2L1cos(2θe−2θ^)−2L1sin(2θe−2θ^)⎦⎤=⎣⎡2(Lq−Ld)sinθ~cosθ~−(Lq−Ld)(1−2sin2θ~)−(Lq−Ld)(1−2sin2θ~)−2(Lq−Ld)sinθ~cosθ~⎦⎤=⎣⎡−L1sin(2θe−2θ^)L0+L1cos(2θe−2θ^)−L0+L1cos(2θe−2θ^)L1sin(2θe−2θ^)⎦⎤=[−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~−Ld−(Lq−Ld)sin2θ~(Lq−Ld)sinθ~cosθ~]=⎣⎡L0−L1cos(2θe−2θ^)−L1sin(2θe−2θ^)−L1sin(2θe−2θ^)L0+L1cos(2θe−2θ^)⎦⎤=[Ld+(Lq−Ld)sin2θ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~]=⎣⎡−sin(θe−θ^)cos(θe−θ^)⎦⎤=[−sinθ~cosθ~]
电压方程可写为
[uγuδ]=A[iγiδ]+Bωe[iγiδ]+Cω^e[iγiδ]+Dddt[iγiδ]+Eωeψf\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =A\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +B\omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +C\hat{\omega}_e \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +D\frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +E\omega _e\psi _f [uγuδ]=A[iγiδ]+Bωe[iγiδ]+Cω^e[iγiδ]+Ddtd[iγiδ]+Eωeψf
带入化简结果，可得：
[uγuδ]=[R00R][iγiδ]+[2(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)(1−2sin⁡2θ~)−(Lq−Ld)(1−2sin⁡2θ~)−2(Lq−Ld)sin⁡θ~cos⁡θ~]ωe[iγiδ]+[−(Lq−Ld)sin⁡θ~cos⁡θ~−Ld−(Lq−Ld)sin⁡2θ~Lq−(Lq−Ld)sin⁡2θ~(Lq−Ld)sin⁡θ~cos⁡θ~]ω^[iγiδ]+[Ld+(Lq−Ld)sin⁡2θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡θ~cos⁡θ~Lq−(Lq−Ld)sin⁡2θ~]ddt[iγiδ]+[−sin⁡θ~cos⁡θ~]ωeψf\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =\left[ \begin{matrix} R& 0\\ 0& R\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} 2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -\left( L_q-L_d \right) \left( 1-2\sin ^2\tilde{\theta} \right)\\ -\left( L_q-L_d \right) \left( 1-2\sin ^2\tilde{\theta} \right)& -2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -L_d-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}& \left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \hat{\omega}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} L_d+\left( L_q-L_d \right) \sin ^2\tilde{\theta}& -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ \end{matrix} \right] \frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{array}{c} -\sin \tilde{\theta}\\ \cos \tilde{\theta}\\ \end{array} \right] \omega _e\psi _f \\ [uγuδ]=[R00R][iγiδ]+⎣⎡2(Lq−Ld)sinθ~cosθ~−(Lq−Ld)(1−2sin2θ~)−(Lq−Ld)(1−2sin2θ~)−2(Lq−Ld)sinθ~cosθ~⎦⎤ωe[iγiδ]+[−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~−Ld−(Lq−Ld)sin2θ~(Lq−Ld)sinθ~cosθ~]ω^[iγiδ]+[Ld+(Lq−Ld)sin2θ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~]dtd[iγiδ]+[−sinθ~cosθ~]ωeψf

匹配第一项

从这里开始，公式的模样开始向参考文献中的样子靠拢，论文中的公式中等号右边共五项。整理上式等式右边的第一项、第四项，可得：
[uγuδ]=[R+pLd−ωeLqωeLdR+pLq][iγiδ]+[0Lq−Ld0]ωe[iγiδ]+[2(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)(1−2sin⁡2θ~)−(Lq−Ld)(1−2sin⁡2θ~)−2(Lq−Ld)sin⁡θ~cos⁡θ~]ωe[iγiδ]+[−(Lq−Ld)sin⁡θ~cos⁡θ~−Ld−(Lq−Ld)sin⁡2θ~Lq−(Lq−Ld)sin⁡2θ~(Lq−Ld)sin⁡θ~cos⁡θ~]ω^e[iγiδ]+[(Lq−Ld)sin⁡2θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡2θ~]ddt[iγiδ]+[−sin⁡θ~cos⁡θ~]ωeψf\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =\left[ \begin{matrix} R+pL_d& -\omega _eL_q\\ \omega _eL_d& R+pL_q\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} 0& L_q\\ -L_d& 0\\ \end{matrix} \right] \omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} 2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -\left( L_q-L_d \right) \left( 1-2\sin ^2\tilde{\theta} \right)\\ -\left( L_q-L_d \right) \left( 1-2\sin ^2\tilde{\theta} \right)& -2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -L_d-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}& \left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right]\hat{\omega}_e \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} \left( L_q-L_d \right) \sin ^2\tilde{\theta}& -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ \end{matrix} \right] \frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{array}{c} -\sin \tilde{\theta}\\ \cos \tilde{\theta}\\ \end{array} \right] \omega _e\psi _f \\ [uγuδ]=[R+pLdωeLd−ωeLqR+pLq][iγiδ]+[0−LdLq0]ωe[iγiδ]+⎣⎡2(Lq−Ld)sinθ~cosθ~−(Lq−Ld)(1−2sin2θ~)−(Lq−Ld)(1−2sin2θ~)−2(Lq−Ld)sinθ~cosθ~⎦⎤ωe[iγiδ]+[−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~−Ld−(Lq−Ld)sin2θ~(Lq−Ld)sinθ~cosθ~]ω^e[iγiδ]+[(Lq−Ld)sin2θ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sin2θ~]dtd[iγiδ]+[−sinθ~cosθ~]ωeψf
注：上式中，等式右边第一项的次对角线与等式右边第二项可抵消，虚构第二项的目的如前所述，从第一项开始，公式逐步与文献中的公式匹配。

匹配第二项、第三项

上式中的第六项（最后一项）移至第二项，第五项（电流微分项）移至第三项
[uγuδ]=[R+pLd−ωeLqωeLdR+pLq][iγiδ]+[−sin⁡θ~cos⁡θ~]ωeψf+[(Lq−Ld)sin⁡2θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡2θ~]ddt[iγiδ]+[2(Lq−Ld)sin⁡θ~cos⁡θ~Lq−(Lq−Ld)+(Lq−Ld)2sin⁡2θ~−Ld−(Lq−Ld)+(Lq−Ld)2sin⁡2θ~−2(Lq−Ld)sin⁡θ~cos⁡θ~]ωe[iγiδ]+[−(Lq−Ld)sin⁡θ~cos⁡θ~−Ld−(Lq−Ld)sin⁡2θ~Lq−(Lq−Ld)sin⁡2θ~(Lq−Ld)sin⁡θ~cos⁡θ~]ω^e[iγiδ]\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =\left[ \begin{matrix} R+pL_d& -\omega _eL_q\\ \omega _eL_d& R+pL_q\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{array}{c} -\sin \tilde{\theta}\\ \cos \tilde{\theta}\\ \end{array} \right] \omega _e\psi _f+\left[ \begin{matrix} \left( L_q-L_d \right) \sin ^2\tilde{\theta}& -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ \end{matrix} \right] \frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} 2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& L_q-\left( L_q-L_d \right) +\left( L_q-L_d \right) 2\sin ^2\tilde{\theta}\\ -L_d-\left( L_q-L_d \right) +\left( L_q-L_d \right) 2\sin ^2\tilde{\theta}& -2\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -L_d-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}& \left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right]\hat{\omega}_e \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \\ [uγuδ]=[R+pLdωeLd−ωeLqR+pLq][iγiδ]+[−sinθ~cosθ~]ωeψf+[(Lq−Ld)sin2θ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sin2θ~]dtd[iγiδ]+[2(Lq−Ld)sinθ~cosθ~−Ld−(Lq−Ld)+(Lq−Ld)2sin2θ~Lq−(Lq−Ld)+(Lq−Ld)2sin2θ~−2(Lq−Ld)sinθ~cosθ~]ωe[iγiδ]+[−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~−Ld−(Lq−Ld)sin2θ~(Lq−Ld)sinθ~cosθ~]ω^e[iγiδ]

匹配第五项、第四项

此步中，观察论文中的公式样子，缺什么补什么即可，最后，整个公式可写为
[uγuδ]=[R+pLd−ωeLqωeLdR+pLq][iγiδ]+[−sin⁡θ~cos⁡θ~]ωeψf+[(Lq−Ld)sin⁡2θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡θ~cos⁡θ~−(Lq−Ld)sin⁡2θ~]ddt[iγiδ]+[−(Lq−Ld)sin⁡θ~cos⁡θ~−Ld−(Lq−Ld)sin⁡2θ~Lq−(Lq−Ld)sin⁡2θ~(Lq−Ld)sin⁡θ~cos⁡θ~](ω^e−ωe)[iγiδ]+[(Lq−Ld)sin⁡θ~cos⁡θ~(Lq−Ld)sin⁡2θ~(Lq−Ld)sin⁡2θ~−(Lq−Ld)sin⁡θ~cos⁡θ~]ωe[iγiδ]\left[ \begin{array}{c} u_{\gamma}\\ u_{\delta}\\ \end{array} \right] =\left[ \begin{matrix} R+pL_d& -\omega _eL_q\\ \omega _eL_d& R+pL_q\\ \end{matrix} \right] \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{array}{c} -\sin \tilde{\theta}\\ \cos \tilde{\theta}\\ \end{array} \right] \omega _e\psi _f+\left[ \begin{matrix} \left( L_q-L_d \right) \sin ^2\tilde{\theta}& -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ \end{matrix} \right] \frac{\mathrm{d}}{\mathrm{dt}}\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& -L_d-\left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ L_q-\left( L_q-L_d \right) \sin ^2\tilde{\theta}& \left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \left( \hat{\omega}_e-\omega _e \right) \left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] +\left[ \begin{matrix} \left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}& \left( L_q-L_d \right) \sin ^2\tilde{\theta}\\ \left( L_q-L_d \right) \sin ^2\tilde{\theta}& -\left( L_q-L_d \right) \sin \tilde{\theta}\cos \tilde{\theta}\\ \end{matrix} \right] \omega _e\left[ \begin{array}{c} i_{\gamma}\\ i_{\delta}\\ \end{array} \right] \\ [uγuδ]=[R+pLdωeLd−ωeLqR+pLq][iγiδ]+[−sinθ~cosθ~]ωeψf+[(Lq−Ld)sin2θ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sinθ~cosθ~−(Lq−Ld)sin2θ~]dtd[iγiδ]+[−(Lq−Ld)sinθ~cosθ~Lq−(Lq−Ld)sin2θ~−Ld−(Lq−Ld)sin2θ~(Lq−Ld)sinθ~cosθ~](ω^e−ωe)[iγiδ]+[(Lq−Ld)sinθ~cosθ~(Lq−Ld)sin2θ~(Lq−Ld)sin2θ~−(Lq−Ld)sinθ~cosθ~]ωe[iγiδ]
至此，上式与论文中估计轴系的扩展反电势公式一致。

参考文献

【1】Morimoto S, Kawamoto K, Sanada M, et al. Sensorless control strategy for salient-pole PMSM based on extended EMF in rotating reference frame[J]. IEEE transactions on industry applications, 2002, 38(4): 1054-1061.

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