Induction Machine (Squirrel Cage)

Purpose

Non-saturable induction machine with squirrel-cage rotor

Library

Electrical / Machines

Description

The machine model is based on a stationary reference frame (Clarke transformation). A sophisticated implementation of the Clarke transformation facilitates the connection of external inductances in series with the stator windings.

The machine operates as a motor or generator; if the mechanical torque has the same sign as the rotational speed the machine is operating in motor mode, otherwise in generator mode. All electrical variables and parameters are viewed from the stator side. In the component icon, phase a of the stator winding is marked with a dot.

In order to inspect the implementation, please select the component in your circuit and choose Look under mask from the Edit > Subsystem menu or the block’s context menu.

Electrical System

Fig. 227 d-axis

Fig. 228 q-axis

The rotor flux is computed as:

\[\Psi_\mathrm{r,d} = L'_\mathrm{lr} \, i'_\mathrm{r,d} + L_\mathrm{m} \, ( i_\mathrm{s,d} + i'_\mathrm{r,d} )\]

\[\Psi_\mathrm{r,q} = L'_\mathrm{lr} \, i'_\mathrm{r,q} + L_\mathrm{m} \, ( i_\mathrm{s,q} + i'_\mathrm{r,q} )\]

The three-phase voltages \(v_\mathrm{s,ab}\) and \(v_\mathrm{s,bc}\) at the stator terminals are transformed into dq quantities:

\[\begin{split}\begin{bmatrix} v_\mathrm{s,d}\\ v_\mathrm{s,q}\\ \end{bmatrix} = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ 0 & \frac{1}{\sqrt{3}} \end{bmatrix} \begin{bmatrix} v_\mathrm{s,ab} \\ v_\mathrm{s,bc} \end{bmatrix}\end{split}\]

Likewise, the stator currents in the stationary reference frame are transformed back into three-phase currents:

\[\begin{split}\begin{bmatrix} i_\mathrm{s,a} \\ i_\mathrm{s,b} \\ i_\mathrm{s,c} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} i_\mathrm{s,d} \\ i_\mathrm{s,q} \end{bmatrix}\end{split}\]

Electro-Mechanical System

Electromagnetic torque:

\[T_\mathrm{e} = \frac{3}{2} \, p \, L_\mathrm{m} ( i_\mathrm{s,q} \, i'_\mathrm{r,d} - i_\mathrm{s,d} \, i'_\mathrm{r,q} )\]

Mechanical System

Mechanical rotor speed \(\omega_\mathrm{m}\):

\[\dot{\omega}_\mathrm{m} = \frac{1}{J} ( T_\mathrm{e} - F \omega_\mathrm{m} - T_\mathrm{m} )\]

\[\omega_\mathrm{e} = p \, \omega_\mathrm{m}\]

Mechanical rotor angle \(\theta_\mathrm{m}\):

\[\dot{\theta}_\mathrm{m} = \omega_\mathrm{m}\]

\[\theta_\mathrm{e} = p \, \theta_\mathrm{m}\]

Parameters

Same as for the Induction Machine (Slip Ring).

Probe Signals

Most probe signals for the Induction Machine (Slip Ring) are also available with this squirrel-cage machine. Only the following probe signal is different:

Rotor currents

The rotor currents \(i'_\mathrm{r,d}\) and \(i'_\mathrm{r,q}\) in the stationary reference frame in amperes \((\mathrm{A})\), referred to the stator side.