Synchronous machine

$$ \left[\begin{array}{c} \psi_{s d} \\ \psi_{s q} \\ \psi_{s 0} \\ \psi_{exc} \\ \psi_{k d_{1}} \\ \psi_{k d_{2}} \\ \psi_{k q_{1}} \\ \psi_{k q_{2}} \\ \psi_{k q_{3}} \end{array}\right]= L \left[\begin{array}{c} i_{s d} \\ i_{s q} \\ i_{s 0} \\ i_{exc} \\ i_{k d_{1}} \\ i_{k d_{2}} \\ i_{k q_{1}} \\ i_{kq_{2}} \\ i_{k q_{3}} \end{array}\right] $$ $$ L=\left[\begin{array}{cc} -L_{d q 0} & L_{a d q 0} \\ -L_{a d q} ^T & L_{k}\end{array}\right] $$ $$ \left\{ \begin{array}{cc} L_{d}=L_{m d}+L_{\sigma s} \\ L_{q}=L_{m q}+L_{\sigma s} \\ \end{array} \right. $$ $$ L_{d q} 0=\left[\begin{array}{ccc}L_{d} & 0 & 0 \\0 & L_{q} & 0 \\0 & 0 & L_{\sigma s}\end{array}\right] $$ $$ L_{\text {adq0}} =\left[\begin{array}{cccccc}L_{m d} & L_{m d} & L_{m d} & 0 & 0 & 0 \\0 & 0 & 0 & L_{m q} & L_{q q} & L_{m q} \\0 & 0 & 0 & 0 & 0 & 0\end{array}\right] $$ $$ \begin{align} L_k = &\left[\begin{array}{cccccc} L_{m d} & L_{m d} & L_{m d} & 0 & 0 & 0 \\ L_{m d} & L_{m d} & L_{m d} & 0 & 0 & 0 \\ L_{m d} & L_{m d} & L_{m d} & 0 & 0 & 0 \\ 0 & 0 & 0 & L_{m q} & L_{m q} & L_{m q} \\ 0 & 0 & 0 & L_{m q} & L_{m q} & L_{m q} \\ 0 & 0 & 0 & L_{m q} & L_{m q} & L_{m q} \end{array}\right] \nonumber \\ +& \left[\begin{array}{cccccc} L_{\sigma exc} & & & & & \\ & L_{k d_{1}} & & & & \\ & & L_{k d_{2}} & & & \\ & & & L_{k q_{1}} & & \\ & & & & L_{kq_{2}} & \\ & & & & & L_{k q_{3}} \end{array}\right] \end{align} $$ The classical synchronous machine swing equation is shown below: $$ \begin{equation} \begin{array}{ccl} T_m - T_e & = & 2H\cdot \dfrac{\mathrm{d}\omega_r}{\mathrm{d}t} \\ \dfrac{\mathrm{d}\delta_r}{\mathrm{d}t} & = & \omega_b \cdot (\omega_r - \omega_{ref}) \end{array} \end{equation} $$