Consider the model of a PMSM motor with non-salient poles:

(1)   \begin{equation*}\left\{\begin{array}{rcl}L \frac{di_{a}}{dt} & = & v_{a}-R i_{a}+ p \phi_f \Omega \sin(p\theta)\\\\L \frac{di_{b}}{dt} & = & v_{b}-R i_{b}+ p \phi_f \Omega \sin\left(p\theta-\frac{2\pi}{3}\right)\\\\L \frac{di_{c}}{dt} & = & v_{c}-R i_{c}+ p \phi_f \Omega \sin\left(p\theta+\frac{2\pi}{3}\right)\\\\J\frac{d\Omega}{dt} &=& \tau_m-\tau_l \\\\{\frac{d\theta}{dt}} &=& \Omega\\\end{array}\right.\end{equation*}

where i_{abc} = \left[\begin{matrix}i_a & i_b & i_c\end{matrix}\right]^\intercal are the phase current, v_{abc} = \left[\begin{matrix}v_a & v_b & v_c\end{matrix}\right]^\intercal the phase voltages, \Omega and \theta are the mechanical rotor speed and position and p the pole pair number. L et R are the phase inductance and resistor, \phi_f is the flux constant generated by permanent magnets. \tau_m is the motor torque and \tau_l the load torque.

(2)   \begin{equation*}\tau_m = -p\phi_f \left[i_a\sin(p\theta) + i_b\sin\left(p\theta-\frac{2\pi}{3}\right) + i_c\sin\left(p\theta+\frac{2\pi}{3}\right)\right]\end{equation*}


PMSM Scheme

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