Doubly Fed Induction Motor

Schematic

../../../_images/ESB_DFIM_alphabeta.svg

Electrical ODE

\[\begin{split}\frac{\mathrm{d} i_\mathrm{s \alpha}}{\mathrm{d} t}&= -\frac{1}{\tau_\sigma} i_\mathrm{s \alpha} + \frac{R_\mathrm{r} L_\mathrm{m}}{\sigma L_\mathrm{r}^2 L_\mathrm{s}} \psi_\mathrm{r \alpha} + p \omega_\mathrm{\text{me}} \frac{L_\mathrm{m}}{\sigma L_\mathrm{r} L_\mathrm{s}} \psi_\mathrm{r \beta} + \frac{1}{\sigma L_\mathrm{s}} u_\mathrm{s \alpha} - \frac{L_\mathrm{m}}{\sigma L_\mathrm{r} L_\mathrm{s}} u_\mathrm{r \alpha}\\ \frac{\mathrm{d} i_\mathrm{s \beta}}{\mathrm{d} t}&= -\frac{1}{\tau_\sigma} i_\mathrm{s \beta} - p \omega_\mathrm{\text{me}} \frac{L_\mathrm{m}}{\sigma L_\mathrm{r} L_\mathrm{s}} \psi_\mathrm{r \alpha} + \frac{R_\mathrm{r} L_\mathrm{m}}{\sigma L_\mathrm{r}^2 L_\mathrm{s}} \psi_\mathrm{r \beta} + \frac{1}{\sigma L_\mathrm{s}} u_\mathrm{s \beta} - \frac{L_\mathrm{m}}{\sigma L_\mathrm{r} L_\mathrm{s}} u_\mathrm{r \beta}\\ \frac{\mathrm{d} \psi_\mathrm{r \alpha}}{\mathrm{d} t}&= \frac{L_\mathrm{m}}{\tau_\mathrm{r}} i_\mathrm{s \alpha} - \frac{1}{\tau_\mathrm{r}} \psi_\mathrm{r \alpha} - p \omega_\mathrm{\text{me}} \psi_\mathrm{r \beta} + u_\mathrm{r \alpha}\\ \frac{\mathrm{d} \psi_\mathrm{r \beta}}{\mathrm{d} t}&= \frac{L_\mathrm{m}}{\tau_\mathrm{r}} i_\mathrm{s \beta} + p \omega_\mathrm{\text{me}} \psi_\mathrm{r \alpha} - \frac{1}{\tau_\mathrm{r}} \psi_\mathrm{r \beta} + u_\mathrm{r \beta}\\ \frac{\mathrm{d} \varepsilon_\mathrm{el}}{\mathrm{d} t}&= p \omega_\mathrm{me}\end{split}\]

with

\[\begin{split}L_\mathrm{s} &= L_\mathrm{m} + L_\mathrm{\sigma s} & \quad L_\mathrm{r} &= L_\mathrm{m} + L_\mathrm{\sigma r}\\ \sigma &= \frac{L_\mathrm{r} L_\mathrm{s} - L_\mathrm{m}^2}{L_\mathrm{r} L_\mathrm{s}} & \quad \tau_\mathrm{r} &=\frac{L_\mathrm{r}}{R_\mathrm{r}} & \quad \tau_\sigma &= \frac{\sigma L_\mathrm{s}}{R_\mathrm{s} + R_\mathrm{r} \frac{L_\mathrm{m}^2}{L_\mathrm{r}^2}}\end{split}\]

Torque Equation

\[T=\frac{3}{2} p \frac{L_\mathrm{m}}{L_\mathrm{r}} (\psi_\mathrm{r \alpha} i_\mathrm{s \beta} - \psi_\mathrm{r \beta} i_\mathrm{s \alpha})\]

Code Documentation