1.1.1 Stationary Reference Frame Model

The following equation represents the stationary model of PMSM:

${\displaystyle \left[\begin{array}{c}{U}_{\alpha }\\ U_{\beta }\end{array}\right]=R_s\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+{p\mathbf{L}}_{\alpha \mathbf{\beta }}\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+\left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]}$

Where:

• ${\displaystyle \left[\begin{array}{c}{U}_{\alpha }\\ U_{\beta }\end{array}\right]}$ is the stator voltage in a stationary reference frame
• ${\displaystyle \left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]}$ is the stator current in a stationary reference frame
• ${\displaystyle \left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]}$ is the stator back EMF in a stationary reference frame
• R_s is the stator per phase resistance
• p = d/dt is the differential operator

${\mathbf{L}}_{\alpha \beta }=[\begin{array}{c}\frac{L_{d+}{L}_q}{2}+\frac{L_d-L_q}{2}\cos \left({2\theta }_e\right)\frac{L_d-L_q}{2}\sin \left(2{\theta }_e\right)\\ \frac{L_d-L_q}{2}\sin \left({2\theta }_e\right)\frac{L_d+L_q}{2}-\frac{L_d-L_q}{2}\cos \left(2{\theta }_e\right)\end{array}]$

• L_d is the inductance when the winding axis is aligned to the rotor flux axis
• L_q is the inductance when the winding axis is orthogonal to the rotor flux axis
• θ_e is the rotor flux angle

${\displaystyle \left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]=w_e{\lambda }_m\left[\begin{array}{c}-\sin \left(2{\theta }_e\right)\\ \cos \left(2{\theta }_e\right)\end{array}\right]}$

• λ_m is the permanent magnet flux linkage
• w_e is the electrical speed of the motor