PMSM Model

Transformation

abc to α-β / 3s-2s

$$ \begin{aligned} \begin{bmatrix} f_{\alpha} \\ f_{\beta} \\ f_0 \\ \end{bmatrix} &= \frac{2}{3} \begin{bmatrix*} 1 & -1/2 & -1/2 \\ 0 & \sqrt{3}/2 & -\sqrt{3}/2 \\ 1/\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} \\ \end{bmatrix*} \begin{bmatrix} f_a \\ f_b \\ f_c \\ \end{bmatrix} = \pmb{T}_{\text{3s-2s}} \begin{bmatrix} f_a \\ f_b \\ f_c \\ \end{bmatrix} \\ \begin{bmatrix} f_a \\ f_b \\ f_c \\ \end{bmatrix} &= \phantom{-} \begin{bmatrix*} 1 & 0 & 1/\sqrt{2} \\ -1/2 & \sqrt{3}/2 & 1/\sqrt{2} \\ -1/2 & -\sqrt{3}/2 & 1/\sqrt{2} \\ \end{bmatrix*} \begin{bmatrix} f_{\alpha} \\ f_{\beta} \\ f_0 \\ \end{bmatrix} =\pmb{T}_{\text{3s-2s}}^{-1}\begin{bmatrix} f_{\alpha} \\ f_{\beta} \\ f_0 \\ \end{bmatrix} \end{aligned} $$

α-β to d-q / 2s-2r

$$ \begin{aligned} {\begin{bmatrix} f_d \\ f_q \\ \end{bmatrix}} &= {\begin{bmatrix*} \phantom{-}\cos\theta_e & \phantom{-}\sin\theta_e \\ -\sin\theta_e & \phantom{-}\cos\theta_e\\ \end{bmatrix*}} {\begin{bmatrix} f_\alpha \\ f_\beta \end{bmatrix}} ={\pmb{T}_{\text{2s-2r}}} {\begin{bmatrix} f_\alpha \\ f_\beta \end{bmatrix}}\\ {\begin{bmatrix} f_\alpha \\ f_\beta \end{bmatrix}} &= {\begin{bmatrix*} \phantom{-}\cos\theta_e & -\sin\theta_e \\ \phantom{-}\sin\theta_e & \phantom{-}\cos\theta_e\\ \end{bmatrix*}} {\begin{bmatrix} f_d \\ f_q \end{bmatrix}} = {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} f_d \\ f_q \end{bmatrix} } \end{aligned} $$

a-b-c

Voltage

$$ \begin{aligned} {\pmb{u_s}} &= \pmb{R_s i_s}+ \frac{d}{dt} \pmb{\it{\Psi_s}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix} u_a \\ u_b \\ u_c \\ \end{bmatrix}} &= {\begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \\ \end{bmatrix}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} +p {\begin{bmatrix} \it{\Psi}_a \\ \it{\Psi}_b \\ \it{\Psi}_c \\ \end{bmatrix}} \\ &= {\begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \\ \end{bmatrix}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} +p {\begin{Bmatrix} {\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} + {\begin{bmatrix} \it{\Psi}_{fa} \\ \it{\Psi}_{fb} \\ \it{\Psi}_{fc} \\ \end{bmatrix}} \end{Bmatrix}} \\ \end{aligned} $$

Flux Leakage

$$ \begin{aligned} {\pmb{\it{\Psi_s}}}={\pmb{L_s i_s}}+{\pmb{\it{\Psi_f}}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix} \it{\Psi}_a \\ \it{\Psi}_b \\ \it{\Psi}_c \\ \end{bmatrix}} &= {\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} + {\begin{bmatrix} \it{\Psi}_{fa} \\ \it{\Psi}_{fb} \\ \it{\Psi}_{fc} \\ \end{bmatrix}} \\ &= {\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos(\theta_e) \\ \cos(\theta_e-2\pi/3) \\ \cos(\theta_e+2\pi/3) \\ \end{bmatrix*}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}} &= {\begin{bmatrix*}[l] \phantom{-}L_{s0}\;+L_{s2}\cos(2(\theta_e)) & -M_{s0}+L_{s2}\cos(2(\theta_e-\pi/3)) & -M_{s0}+L_{s2}\cos(2(\theta_e+\pi/3)) \\ -M_{s0}+L_{s2}\cos(2(\theta_e-\pi/3)) & \phantom{-}L_{s0}\;+L_{s2}\cos(2(\theta_e-2\pi/3)) & -M_{s0}+L_{s2}\cos(2(\theta_e)) \\ -M_{s0}+L_{s2}\cos(2(\theta_e+\pi/3)) & -M_{s0}+L_{s2}\cos(2(\theta_e)) &\phantom{-}L_{s0}\;+L_{s2}\cos(2(\theta_e+2\pi/3)) \\ \end{bmatrix*}} \end{aligned} $$

Torque

$$ \begin{aligned} T_e=\frac{P_e}{\omega_m} \end{aligned} $$

Mechanical motion

$$ \begin{aligned} J \frac{d\omega_m}{dt}&=T_e-T_L-B \omega_m \\ \end{aligned} $$$$ \begin{aligned} \omega_e =p_n \omega_m \end{aligned} $$

α-β

a-b-c to α-β

$$ \begin{aligned} {\begin{bmatrix} u_a \\ u_b \\ u_c \\ \end{bmatrix}} &= {\begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \\ \end{bmatrix}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} +p {\begin{Bmatrix} {{\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}}} {\begin{bmatrix} i_a \\ i_b \\ i_c \\ \end{bmatrix}} +{\it\Psi_f} {\begin{bmatrix*}[l] \cos(\theta_e) \\ \cos(\theta_e-2\pi/3) \\ \cos(\theta_e+2\pi/3) \\ \end{bmatrix*}} \end{Bmatrix}} \\ {\pmb{T}_{\text{3s-2s}}^{-1}\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} &= {\begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \\ \end{bmatrix}} {\pmb{T}_{\text{3s-2s}}^{-1}\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} +p {\begin{Bmatrix} {{\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}}} \pmb{T}_{\text{3s-2s}}^{-1}{\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos(\theta_e) \\ \cos(\theta_e-2\pi/3) \\ \cos(\theta_e+2\pi/3) \\ \end{bmatrix*}} \end{Bmatrix}} \\ {\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} &= \pmb{T}_{\text{3s-2s}} {\begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \\ \end{bmatrix}} {\pmb{T}_{\text{3s-2s}}^{-1}\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} + \pmb{T}_{\text{3s-2s}}\;p {\begin{Bmatrix} {{\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}}} \pmb{T}_{\text{3s-2s}}^{-1}{\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} +{\it\Psi_f} {\begin{bmatrix*}[l] \cos(\theta_e) \\ \cos(\theta_e-2\pi/3) \\ \cos(\theta_e+2\pi/3) \\ \end{bmatrix*}} \end{Bmatrix}} \\ \end{aligned} $$$$ \begin{aligned} &{\pmb{T}_{\text{3s-2s}}} {\begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \\ \end{bmatrix}} {\pmb{T}_{\text{3s-2s}}^{-1}} = {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} \\ &{\pmb{T}_{\text{3s-2s}}} {{\begin{bmatrix} L_{aa} & M_{ab} & M_{ac} \\ M_{ba} & L_{bb} & M_{bc} \\ M_{ca} & M_{cc} & L_{cc} \\ \end{bmatrix}}} {\pmb{T}_{\text{3s-2s}}^{-1}} = {\begin{bmatrix*}[r] L_{s0}+M_{s0}+3/2\;L_{s2}\cos(2(\theta_e)) & 3/2\;L_{s2}\sin(2(\theta_e)) \\ 3/2\;L_{s2}\sin(2(\theta_e)) & L_{s0}+M_{s0}-3/2\;L_{s2}\cos(2(\theta_e)) \\ \end{bmatrix*}} \\ &{\pmb{T}_{\text{3s-2s}}} {\begin{bmatrix*}[l] \cos(\theta_e) \\ \cos(\theta_e-2\pi/3) \\ \cos(\theta_e+2\pi/3) \\ \end{bmatrix*}} = {\begin{bmatrix*}[l] \cos(\theta_e) \\ \sin(\theta_e) \end{bmatrix*}} \end{aligned} $$

α-β Voltage

$$ \begin{aligned} {\pmb{u_{\alpha\beta}}} &= \pmb{R_s i_{\alpha\beta}}+ \frac{d}{dt} \pmb{\it{\Psi_{\alpha\beta}}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} &= {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\begin{bmatrix} i_\alpha \\ i_\beta \\ \end{bmatrix}} + p {\begin{bmatrix*}[l] {\it\Psi_{\alpha}} \\ {\it\Psi_{\beta}} \\ \end{bmatrix*}} \\ &= {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\begin{bmatrix} i_\alpha \\ i_\beta \\ \end{bmatrix}} + p {\begin{Bmatrix} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} + {\begin{bmatrix*}[l] {\it\Psi_{f\alpha}} \\ {\it\Psi_{f\beta}} \\ \end{bmatrix*}} \end{Bmatrix}} \\ &= {\begin{bmatrix*}[r] R_s+pL_{\alpha\alpha} & p M_{\alpha\beta}\\ p M_{\beta\alpha} & R_s +pL_{\beta\beta} \end{bmatrix*}} {\begin{bmatrix} i_\alpha \\ i_\beta \\ \end{bmatrix}} + {\omega_e{\it\Psi_f}} {\begin{bmatrix*}[r] -\sin\theta_e \\ \cos\theta_e \\ \end{bmatrix*}} \\ \end{aligned} $$

α-β Flux Leakage

$$ \begin{aligned} {\pmb{\it{\Psi_{\alpha\beta}}}}={\pmb{L_{\alpha\beta} i_{\alpha\beta}}}+{\pmb{\it{\Psi_{f\alpha\beta}}}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix*}[l] {\it\Psi_{\alpha}} \\ {\it\Psi_{\beta}} \\ \end{bmatrix*}} &= {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} + {\begin{bmatrix*}[l] {\it\Psi_{f\alpha}} \\ {\it\Psi_{f\beta}} \\ \end{bmatrix*}} \\ &= {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{aligned} $$$$ \begin{aligned} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} &= {\begin{bmatrix*}[r] L_{s0}+M_{s0}+3/2\;L_{s2}\cos(2(\theta_e)) & 3/2\;L_{s2}\sin(2(\theta_e)) \\ 3/2\;L_{s2}\sin(2(\theta_e)) & L_{s0}+M_{s0}-3/2\;L_{s2}\cos(2(\theta_e)) \\ \end{bmatrix*}} \\ &= {\begin{bmatrix*}[r] (L_d+L_q)/2+(L_d-L_q)/2\;\cos(2(\theta_e)) & (L_d-L_q)/2\;\sin(2(\theta_e)) \\ (L_d-L_q)/2\;\sin(2(\theta_e)) & (L_d+L_q)/2-(L_d-L_q)/2\;\cos(2(\theta_e)) \\ \end{bmatrix*}}\\ &= {\begin{bmatrix*}[r] \Sigma L+\Delta L\;\cos(2(\theta_e)) & \Delta L\;\sin(2(\theta_e)) \\ \Delta L\;\sin(2(\theta_e)) & \Sigma L-\Delta L\;\cos(2(\theta_e)) \\ \end{bmatrix*}} , {\begin{pmatrix} \Sigma L = (L_d+L_q)/2 \\ \Delta L = (L_d-L_q)/2 \end{pmatrix}} \end{aligned} $$

α-β Torque

$$ \begin{aligned} T_e=\frac{3}{2}p_n[\it\Psi_{\alpha} i_{\beta}-\it\Psi_{\beta} i_{\alpha}] \end{aligned} $$

d-q

α-β to d-q

$$ \begin{aligned} {\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} &= {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\begin{bmatrix} i_\alpha \\ i_\beta \\ \end{bmatrix}} + p {\begin{Bmatrix} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{Bmatrix}} \\ {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} u_d \\ u_q \end{bmatrix}} &= {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} +p {\begin{Bmatrix} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{Bmatrix}} \\ {\begin{bmatrix} u_d \\ u_q \end{bmatrix}} &= {\pmb{T}_{\text{2s-2r}}} {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} +{\pmb{T}_{\text{2s-2r}}}\;p {\begin{Bmatrix} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{Bmatrix}} \\ \end{aligned} $$$$ \begin{aligned} &{\pmb{T}_{\text{2s-2r}}} {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\pmb{T}_{\text{2s-2r}}^{-1}} = {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}}\\ &{\pmb{T}_{\text{2s-2r}}} \;p {\begin{Bmatrix} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{Bmatrix}} \\ &= p {\begin{Bmatrix} {\pmb{T}_{\text{2s-2r}}} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\it\Psi_f}\; {\pmb{T}_{\text{2s-2r}}} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{Bmatrix}} - (p\;{\pmb{T}_{\text{2s-2r}}}) {\begin{Bmatrix} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\it\Psi_f} {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} \end{Bmatrix}} \\ \\ &{\pmb{T}_{\text{2s-2r}}} {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} = {\begin{bmatrix*}[c] L_{s0}+M_{s0}+3/2\;L_{s2} & 0 \\ 0 & L_{s0}+M_{s0}-3/2\;L_{s2} \\ \end{bmatrix*}} \\ &(p\;{\pmb{T}_{\text{2s-2r}}}) {{\begin{bmatrix} L_{\alpha\alpha} & M_{\alpha\beta} \\ M_{\beta\alpha} & L_{\beta\beta} \\ \end{bmatrix}}} {\pmb{T}_{\text{2s-2r}}^{-1}} = {\begin{bmatrix} 0 & \omega_e(L_{s0}+M_{s0}-3/2\;L_{s2}) \\ -\omega_e(L_{s0}+M_{s0}+3/2\;L_{s2}) & 0 \\ \end{bmatrix}}\\ &(p\;{\pmb{T}_{\text{2s-2r}}}) {\begin{bmatrix*}[l] \cos\theta_e \\ \sin\theta_e \\ \end{bmatrix*}} = {\omega_e}{\begin{bmatrix} 0 \\ -1 \\ \end{bmatrix}} \end{aligned} $$

d-q Voltage

$$ \begin{aligned} {\pmb{u_{dq}}} &= {\pmb{R_s i_{dq}}}+ {\frac{d}{dt}}{\pmb{\it{\Psi_{dq}}}}+ {\omega_e} {\begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix}} {\pmb{\it{\Psi_{dq}}}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix} u_d \\ u_q \end{bmatrix}} &= {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} +p {\begin{bmatrix*}[l] {\it\Psi_{d}} \\ {\it\Psi_{q}} \\ \end{bmatrix*}} + {\omega_e} {\begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix}} {\begin{bmatrix*}[l] {\it\Psi_{d}} \\ {\it\Psi_{q}} \\ \end{bmatrix*}} \\ &= {\begin{bmatrix} R_s & 0\\ 0 & R_s \end{bmatrix}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} +p {\begin{bmatrix} L_d & 0 \\ 0 & L_q \\ \end{bmatrix}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\begin{bmatrix} 0 & -{\omega_e L_q} \\ {\omega_e L_d} & 0 \\ \end{bmatrix}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\omega_e{\it\Psi_f}} {\begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}} \\ &= {\begin{bmatrix*}[r] R_s+p L_d & -{\omega_e L_q} \\ {\omega_e L_d} & R_s+p L_q \end{bmatrix*}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\omega_e{\it\Psi_f}} {\begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*}} \end{aligned} $$

d-q Flux Leakage

$$ \begin{aligned} {\pmb{\it{\Psi_{dq}}}}={\pmb{L_{dq} i_{dq}}}+{\pmb{\it{\Psi_{fdq}}}} \end{aligned} $$$$ \begin{aligned} {\begin{bmatrix*} {\it\Psi_{d}} \\ {\it\Psi_{q}} \\ \end{bmatrix*}} = {{\begin{bmatrix} L_d & 0 \\ 0 & L_q \\ \end{bmatrix}}} {\begin{bmatrix*}[l] {i_d} \\ {i_q} \\ \end{bmatrix*}}+{\it\Psi_f}{\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}} \end{aligned} $$$$ \begin{aligned} {{\begin{bmatrix} L_d & 0 \\ 0 & L_q \\ \end{bmatrix}}} &= {\begin{bmatrix*}[c] L_{s0}+M_{s0}+3/2\;L_{s2} & 0 \\ 0 & L_{s0}+M_{s0}-3/2\;L_{s2} \\ \end{bmatrix*}} \end{aligned} $$

d-q Torque

$$ \begin{aligned} T_e &= 3/2\; p_n[\it\Psi_d i_q-\it\Psi_q i_d] \\ &= 3/2\; p_n[(L_d-L_q)i_d i_q+\it\Psi_f i_q] \end{aligned} $$$$ \begin{aligned} T_e = \underbrace{3/2\; p_n (L_d-L_q)i_d i_q}_{\text{Reluctance torque}} +\underbrace{3/2\; p_n \it\Psi_f i_q}_{\text{Magnet torque}} \end{aligned} $$