# Motor Control University

## PMSM modeling

The motor considered here is a three-phase, star-connected motor. The motor is assumed to be electrically balanced. Magnetic saturations are not taken into account (inductances are independent of current). Finally, the EMF (Electro-Mechanical Forces) are sinusoidal.

### Operative part scheme

The figure below represents the whole system including the motor and its inverter.

The system can be described with the following bloc diagram.

### Classical \(abc\) frame

The electrical equation in the \(abc\) frame is given by: \[
L\frac{di_{abc}}{dt} = v_{abc}-R i_{abc}- e_{abc},
\] with: \[
e_{abc}= -p\phi_f\omega \begin{bmatrix}\sin(p\theta)\\\sin(p\theta-\frac{2\pi}{3})\\\sin(p\theta+\frac{2\pi}{3})\end{bmatrix},
\] where \(i_{abc} = \left[\begin{matrix}i_a & i_b & i_c\end{matrix}\right]^\intercal\) is the stator phase current, \(v_{abc} = \left[\begin{matrix}v_a & v_b & v_c\end{matrix}\right]^\intercal\) the stator phase voltage, \(\omega\) and \(\theta\) are the speed and position respectively and \(p\) the pole pair number. \(L\) and \(R\) and the stator inductance and resistor, \(\phi_f\) the flux constant.

The mechanical equations are given by: \[
\begin{array}{lcl}
J \frac{d\omega}{dt}&=&\tau_m-f_v\omega -\tau_l,\\
\frac{d\theta}{dt} &=& \omega,
\end{array}
\] where J is the inertia, \(f_v\) is the viscous friction coefficient. The electromechanical torque is denoted \(\tau_m\) given by : \[
\tau_m = \frac{1}{\omega} e_{abc}^\intercal i_{abc}.
\] and \(\tau_l\) the load torque.

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