PMSM EEMF Model

Voltage Equation

Define $(L_d-L_q)(\omega_e i_d -p i_q)+{\omega_e{\it\Psi_f}}$ as extended electromotive force, EEMF.
Rewritten the α-β voltage equation,

$$ \begin{aligned} {\begin{bmatrix} u_\alpha \\ u_\beta \\ \end{bmatrix}} &= {\begin{bmatrix*}[c] R_s+p L_d & \omega_e (L_d-L_q)\\ -\omega_e (L_d-L_q) & R_s +pL_d \end{bmatrix*}} {\begin{bmatrix} i_{\alpha} \\ i_{\beta} \\ \end{bmatrix}} + {\begin{Bmatrix} (L_d-L_q)(\omega_e i_d -p i_q)+{\omega_e{\it\Psi_f}} \end{Bmatrix}} {\begin{bmatrix*}[r] -\sin\theta_e \\ \cos\theta_e \\ \end{bmatrix*}} \end{aligned} $$

Rewritten the d-q voltage equation,

$$ \begin{aligned} {\begin{bmatrix} u_d \\ u_q \end{bmatrix}} &= {\begin{bmatrix*}[r] R_s+p L_d & -{\omega_e L_q} \\ {\omega_e L_q} & R_s+p L_d \end{bmatrix*}} {\begin{bmatrix} i_d \\ i_q \\ \end{bmatrix}} + {\begin{Bmatrix} (L_d-L_q)(\omega_e i_d -p i_q)+{\omega_e{\it\Psi_f}} \end{Bmatrix}} {\begin{bmatrix*}[r] 0 \\ 1 \\ \end{bmatrix*}} \end{aligned} $$

Sensorless Control based on EEMF Observer

Sensorless control based on EEMF observer.

PMSM EEMF Model

Voltage Equation