*2.1. Induction Machine Model*

The dynamic model of the induction machine is important for the study of transient analysis on computers. If the currents in the rotating reference frame are selected as the main variables, then the state space stator voltage equations in the rotating reference frame can be obtained as

$$
\begin{bmatrix} v\_{qs} \\ v\_{ds} \end{bmatrix} = \begin{bmatrix} R\_s + L\_s p & \omega\_c L\_s & L\_m p & \omega\_c L\_m \\ -\omega\_c L\_s & R\_s + L\_s p & \omega\_c L\_m & L\_m p \end{bmatrix} \begin{bmatrix} i\_{qs} \\ i\_{ds} \\ i\_{qr} \\ i\_{dr} \end{bmatrix} \tag{1}
$$

where *vqs*, *vds*, *iqs*, and *ids* are the stator q-axis and d-axis voltages and currents, respectively; *iqr* and *idr* are the rotor q-axis and d-axis currents, respectively; *Rs* and *Ls* the stator resistance and inductance, respectively; *Lm* is the magnetizing inductance; *ω<sup>e</sup>* is the synchronous rotating angular velocity; and *p* is the differential factor.

The rotor flux linkage expressions in terms of the currents can be written as

$$
\begin{bmatrix}
\psi\_{qr} \\
\psi\_{dr}
\end{bmatrix} = \begin{bmatrix}
L\_m & 0 & L\_r + L\_m & 0 \\
0 & L\_m & 0 & L\_r + L\_m
\end{bmatrix} \begin{bmatrix}
i\_{qs} \\
i\_{ds} \\
i\_{qr} \\
i\_{dr}
\end{bmatrix},
\tag{2}
$$

where *ψqr* and *ψdr* are the rotor q-axis and d-axis flux linkages, respectively. *Lr* is the inductance of the rotor.

The d-axis is located on the rotor flux linkage in the IRFOC. Therefore, *ψqr* = 0 and *ψ<sup>r</sup>* = *ψdr*. By Equation (2), the rotor current in Equation (1) can be substituted as

$$
\omega\_{\mathfrak{q}\mathfrak{s}} = R\_s \dot{i}\_{\mathfrak{q}\mathfrak{s}} + \delta L\_s \frac{d \dot{i}\_{\mathfrak{q}\mathfrak{s}}}{dt} + \omega\_\mathfrak{c} L\_s \dot{i}\_{\mathfrak{d}\mathfrak{s}} + \frac{\omega\_\mathfrak{c} L\_m}{L\_r} (\psi\_r - L\_m \dot{i}\_{\mathfrak{d}\mathfrak{s}}), \tag{3}
$$

$$w\_{ds} = R\_s i\_{ds} + L\_s \frac{di\_{ds}}{dt} - \omega\_c \delta L\_s i\_{qs} + \frac{L\_m}{L\_r} \frac{d(\psi\_r - L\_m i\_{ds})}{dt},\tag{4}$$

where *<sup>δ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *m LsLr*

The rotor flux in the IRFOC can be expressed as

$$\frac{L\_r}{R\_r}\frac{d\psi\_r}{dt} + \psi\_r = L\_m \dot{i}\_{ds\prime} \tag{5}$$

Then, Equation (5) is substituted into Equation (3) to obtain

$$w\_{qs} = R\_s i\_{qs} + \delta L\_s \frac{di\_{qs}}{dt} + \omega\_c L\_s i\_{ds} - \frac{\omega\_c L\_m}{R\_r} \frac{d\psi\_r}{dt} \tag{6}$$

And Equation (4) can be given as

.

$$
\omega\_{ds} = R\_s \dot{i}\_{ds} + \delta L\_s \frac{d\dot{i}\_{ds}}{dt} - \omega\_c \delta L\_s \dot{i}\_{qs} + \frac{L\_m}{L\_r} \frac{d\psi\_r}{dt} \tag{7}
$$
