1.1.2 Rotational Reference Frame Model

The rotational reference frame model of the PMSM is derived by linear transformation of the PMSM equations from a stationary reference frame to a rotating reference frame (Park's Transformation). The transformation essentially converts all sinusoidally varying components to two DC components, thereby making control and analysis using classical control straightforward. The first component is aligned to the rotor flux ( d-axis), while the second component is orthogonal to the rotor flux ( q-axis ). The step wise derivation of the model is beyond the scope of this document, and is therefore not discussed.

The following equation shows the final model in the rotational reference frame after the Park's transformation.

${\displaystyle \left[\begin{array}{c}{U}_d\\ U_q\end{array}\right]=R_s\left[\begin{array}{c}{I}_d\\ I_q\end{array}\right]+{p\mathbf{L}}_{\mathbf{d}\mathbf{q}}\left[\begin{array}{c}{I}_d\\ I_q\end{array}\right]+{\mathbf{J}w}_e{\mathbf{L}}_{\mathbf{d}\mathbf{q}}\left[\begin{array}{c}{I}_d\\ I_q\end{array}\right]+w_e\left[\begin{array}{c}0\\ {\lambda }_m\end{array}\right]}$

Where:

• ${\displaystyle \left[\begin{array}{c}{U}_d\\ U_q\end{array}\right]}$ is the stator phase voltage vector in a rotating reference frame
• ${\displaystyle \left[\begin{array}{c}{I}_d\\ I_q\end{array}\right]}$ is the stator phase current vector in a rotating reference frame
• ${\displaystyle \mathbf{J}\mathrm{=}\mathrm{[}\begin{array}{c}\mathrm{0}\\ \mathrm{1}\end{array}\mathrm{}\begin{array}{c}\mathrm{-}\mathrm{1}\\ \mathrm{0}\end{array}]}$ is the rotational matrix

${\displaystyle {\mathbf{L}}_{\mathbf{d}\mathbf{q}}=[\begin{array}{c}{L}_d\\ 0\end{array}\begin{array}{c}0\\ L_q\end{array}]}$

Where:

• L_d is the inductance along the d-axis
• L_q is the inductance along the q-axis