Induction Machine (Open Stator Windings)

Purpose

Non-saturable induction machine with squirrel-cage rotor and open stator windings

Library

Electrical / Machines

Description

This model of a squirrel-cage induction machine can only be used with the continuous state-space method. 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, the positive terminal of phase a of the stator windings 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

The equivalent electrical circuits for the d, q and 0 axes are shown in Fig. 225.

d-axis — Fig. 225 Equivalent electrical circuits of the d, q and 0 axes of an open stator winding machine.

q-axis — Fig. 225 Equivalent electrical circuits of the d, q and 0 axes of an open stator winding machine.

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 \(v_\mathrm{s,a}\), \(v_\mathrm{s,b}\) and \(v_\mathrm{s,c}\) across the individual stator windings are transformed into dq0 quantities:

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

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

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

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

Most parameters for the Induction Machine (Slip Ring) are also applicable for this machine. Only the following parameter differs:

Initial stator currents: A three-element vector containing the initial stator currents \(i_\mathrm{s,a,0}\), \(i_\mathrm{s,b,0}\) and \(i_\mathrm{s,c,0}\) of phase a, b and c in amperes \((\mathrm{A})\).

Probe Signals

Most probe signals for the Induction Machine (Slip Ring) are also available with this 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.