Induction Machine (Slip Ring)

Purpose

Non-saturable induction machine with slip-ring rotor

Library

Electrical / Machines

Description

This model of a slip-ring induction machine can only be used with the continuous state-space method. If you want to use the discrete state-space method or if you need to take saturation into account, please use the Induction Machine with Saturation.

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. However, external inductors cannot be connected to the rotor windings due to the current sources in the model. In this case, external inductors must be included in the leakage inductance of the rotor.

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 and rotor 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. If you want to make changes, you must first choose Break library link and then Unprotect, both from the same menu.

Electrical System

The equivalent electrical circuits for the d and q axes are shown in Fig. 226.

d-axis

q-axis

Fig. 226 Equivalent electrical circuits of the d and q axes of a slip ring induction 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 \(v_\mathrm{s,ab}\) 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:

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

Similar equations apply to the voltages and currents at the rotor terminals with \(\theta\) being the electrical rotor position:

\[\begin{split}\begin{bmatrix} v'_\mathrm{r,d} \\ v'_\mathrm{r,q} \\ \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos(\theta) & -\cos(\theta - \frac{2\pi}{3})\\ \sin(\theta) & -\sin(\theta - \frac{2\pi}{3}) \end{bmatrix} \begin{bmatrix} v'_\mathrm{r,ab} \\ v'_\mathrm{r,bc} \end{bmatrix}\end{split}\]

\[\begin{split}\begin{bmatrix} i'_\mathrm{r,a} \\ i'_\mathrm{r,b} \\ i'_\mathrm{r,c} \end{bmatrix} = \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ \cos(\theta + \frac{2\pi}{3}) & \sin(\theta + \frac{2\pi}{3}) \\ \cos(\theta - \frac{2\pi}{3}) & \sin(\theta - \frac{2\pi}{3}) \end{bmatrix} \begin{bmatrix} i'_\mathrm{r,d} \\ i'_\mathrm{r,q} \end{bmatrix}\end{split}\]

Electro-Mechanical System

Electromagnetic torque:

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

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

Stator resistance

Stator winding resistance \(R_\mathrm{s}\) in ohms \((\Omega)\).

Stator leakage inductance

Stator leakage inductance \(L_\mathrm{ls}\) in henries \((\mathrm{H})\).

Rotor resistance

Rotor winding resistance \(R'_\mathrm{r}\) in ohms \((\Omega)\), referred to the stator side.

Rotor leakage inductance

Rotor leakage inductance \(L'_\mathrm{lr}\) in henries \((\mathrm{H})\), referred to the stator side.

Magnetizing inductance

Magnetizing inductance \(L_\mathrm{m}\) in henries \((\mathrm{H})\), referred to the stator side.

Inertia

Combined rotor and load inertia \(J\) in \((\mathrm{Nms^2})\).

Friction coefficient

Viscous friction \(F\) in \((\mathrm{Nms})\).

Number of pole pairs

Number of pole pairs \(p\).

Initial rotor speed

Initial mechanical rotor speed \(\omega_\mathrm{m,0}\) in \((\frac{\mathrm{rad}}{\mathrm{s}})\).

Initial rotor position

Initial mechanical rotor angle \(\theta_\mathrm{m,0}\) in radians. If \(\theta_\mathrm{m,0}\) is an integer multiple of \(2\pi/p\), the stator windings are aligned with the rotor windings at simulation start.

Initial stator currents

A two-element vector containing the initial stator currents \(i_\mathrm{s,a,0}\) and \(i_\mathrm{s,b,0}\) of phases a and b in amperes \((\mathrm{A})\).

Initial stator flux

A two-element vector containing the initial stator flux \(\Psi'_\mathrm{s,d,0}\) and \(\Psi'_\mathrm{s,q,0}\) in the stationary reference frame in \((\mathrm{Vs})\).

Probe Signals

Stator phase currents

The three-phase stator winding currents \(i_\mathrm{s,a}\), \(i_\mathrm{s,b}\) and \(i_\mathrm{s,c}\), in amperes \((\mathrm{A})\). Currents flowing into the machine are considered positive.

Rotor phase currents

The three-phase rotor winding currents \(i'_\mathrm{r,a}\), \(i'_\mathrm{r,b}\) and \(i'_\mathrm{r,c}\) in amperes \((\mathrm{A})\), referred to the stator side. Currents flowing into the machine are considered positive.

Stator flux (dq)

The stator flux linkages \(\Psi_\mathrm{s,d}\) and \(\Psi_\mathrm{s,q}\) in the stationary reference frame in weber \((\mathrm{Wb})\):

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

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

Magnetizing flux (dq)

The magnetizing flux linkages \(\Psi_\mathrm{m,d}\) and \(\Psi_\mathrm{m,q}\) in the stationary reference frame in \((\mathrm{Vs})\):

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

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

Rotor flux (dq)

The rotor flux linkages \(\Psi'_\mathrm{r,d}\) and \(\Psi'_\mathrm{r,q}\) in the stationary reference frame in \((\mathrm{Vs})\).

Rotational speed

The rotational speed \(\omega_\mathrm{m}\) of the rotor in \((\frac{\mathrm{rad}}{\mathrm{s}})\).

Rotor position

The mechanical rotor angle \(\theta_\mathrm{m}\) in radians.

Electrical torque

The electrical torque \(T_\mathrm{e}\) of the machine in \((\mathrm{Nm})\).