Non-Excited Synchronous Machine

Purpose

Non-excited synchronous machine configurable with lookup tables

Library

Electrical / Machines

Description

This three-phase synchronous machine has a solid rotor with optional permanent magnets. Magnetization, saliency, saturation and cross-coupling are modeled by means of corresponding flux linkage and incremental inductance lookup tables.

The machine can operate as either a motor or generator. If the mechanical torque has the same sign as the rotational speed, the machine is operating in motor mode; otherwise it is in generator mode. In the component icon, phase a 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 model is realized by means of the voltage behind reactance (VBR) formulation and is therefore appropriate to simulate switching dead-time and failure modes.

Electrical equation expressed in synchronous frame by means of space-vector notation:

\[\vec{v}_\mathrm{s} = R_\mathrm{s} \cdot \vec{i}_\mathrm{s} + (\mathbf{L}_{\sigma \mathrm{s}} + \mathbf{L}_{\mathrm{mi}}) \cdot (\frac{d\vec{i}_\mathrm{s}}{dt} + j \cdot \omega \cdot \vec{i}_\mathrm{s}) + \vec{e}_\mathrm{s}\]

\[\vec{e}_\mathrm{s} = j \cdot \omega \cdot \vec{\varphi}_\mathrm{m} - \mathbf{L}_{\mathrm{mi}} \cdot j \cdot \omega \cdot \vec{i}_\mathrm{s}\]

with

\[\begin{split}\vec{x} = \begin{bmatrix} x_\mathrm{d} \\ x_\mathrm{q} \end{bmatrix}\end{split}\]

where \(j \cdot \vec{x}\) rotates the synchronous frame vector, \(\vec{x}\), clockwise by \(90^\circ\).

These equations are transformed back into the stationary frame to control the VBR network. The zero-sequence impedance of the machine is set to Lss.

The VBR network implementation uses the Variable Inductor to model the time-varying characteristic of the machine inductance. PLECS does not support code generation for the Variable Inductor component. When generating code for the VBR model the machine equations are reformulated to interface electrically with a constant inductance and emulate the variable portion of the inductance with a voltage source.

Electro-Mechanical System

Electromagnetic torque:

\[T_\mathrm{e} = \frac{3}{2} \, p \, ( \varphi_\mathrm{d} \, i_\mathrm{q} - \varphi_\mathrm{q} \, i_\mathrm{d} )\]

Mechanical System

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

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

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

Parameters

General

Stator resistance: Armature or stator resistance \(R_\mathrm{s}\) in \(\Omega\).
Stator leakage inductance: Leakage inductance of stator windings in henries \((\mathrm{H})\). Stator leakage must be set to a non-zero value.
Number of pole pairs: Number of pole pairs \(p\).
Initial stator currents: A two-element vector containing the initial stator currents \(i_\mathrm{a,0}\) and \(i_\mathrm{b,0}\) of phase a and b in amperes \((\mathrm{A})\). \(i_\mathrm{c,0}\) is calculated assuming a neutral connection.

Magnetizing Inductance

Id lookup vector

d-axis current vector serving as input values to flux linkage and incremental inductance lookup tables. Must be a vector with \(2\) or more elements, and monotonically increasing, i.e. \([0 \ldots i_\mathrm{d,max}]\). The values are in amperes \((\mathrm{A})\).

Iq lookup vector

q-axis current vector serving as input values to flux linkage and incremental inductance lookup tables. Must be a vector with \(2`\) or more elements, and monotonically increasing, i.e. \([0 \ldots i_\mathrm{q,max}]\). The values are in amperes \((\mathrm{A})\).

Psid (Id, Iq) lookup table

d-axis flux linkage lookup table (2D). The number of rows and columns must match the size of the d- and q-axis currents, respectively. The values are in volt-seconds \((\mathrm{Vs})\).

Psiq (Id, Iq) lookup table

q-axis flux linkage lookup table (2D). The number of rows and columns must match the size of the d- and q-axis currents, respectively. The values are in volt-seconds \((\mathrm{Vs})\).

Lmidd (Id, Iq) lookup table

d-axis self incremental inductance lookup table (2D). The number of rows and columns must match the size of the d- and q-axis currents, respectively. If Lmi data is not specified or set to [], the incremental inductance is calculated from the flux linkage data. The values are in henries \((\mathrm{H})\).

Lmiqq (Id, Iq) lookup table

q-axis self incremental inductance lookup table (2D). The number of rows and columns must match the size of the d- and q-axis currents, respectively. If Lmi data is not specified or set to [], the incremental inductance is calculated from the flux linkage data. The values are in henries \((\mathrm{H})\).

Lmidq (Id, Iq) lookup table

Mutual incremental inductance lookup table (2D). The number of rows and columns must match the size of the d- and q-axis currents, respectively. If Lmi data is not specified or set to [], the incremental inductance is calculated from the flux linkage data. The values are in henries \((\mathrm{H})\).

Generated table size {[d, q]}

User-specified dimension to generate derived current vectors and corresponding flux linkage and incremental inductance lookup tables.

If left empty, the supplied data is used as-is. If specified, the dimensions of the rows and columns must be \(2\) or more.

Specifying a scalar value, n, will generate equally spaced, n-element d- and q-axis current vectors. The corresponding 2D lookup tables for flux linkage and incremental inductance are also generated.

Specifying a vector, [m,n], will generate equally spaced d- and q-axis current vectors. The d-axis current vector will have m elements and the q-axis current vector will have n elements. The corresponding 2D lookup tables for flux linkage and incremental inductance are also generated.

The size of the generated tables affect the model initialization and simulation speeds. A smaller size leads to faster model initialization and simulation speeds, but lower resolution in the generated tables. A larger size increases the resolution but adversely affects the model initialization and simulation speeds. Care must be taken when configuring this parameter.

Current out of range

Configure to ignore, warn, warn and pause simulation, or generate error and stop simulation if the d-axis or q-axis currents are outside the specified range.

Mechanical

Inertia: Combined rotor and load inertia \(J\) in \((\mathrm{Nms^2})\).
Friction coefficient: Viscous friction \(F\) in \((\mathrm{Nms})\).
Initial rotor speed: Initial mechanical rotor speed \(\omega_\mathrm{m,0}\) in radians per second \(\left( \frac{\mathrm{rad}}{\mathrm{s}} \right)\).
Initial rotor position: Initial mechanical rotor angle \(\theta_\mathrm{m,0}\) in radians.

Probe Signals

Stator phase currents: The three-phase stator winding currents \(i_\mathrm{a}\), \(i_\mathrm{b}\) and \(i_\mathrm{c}\), in amperes \((\mathrm{A})\). Currents flowing into the machine are considered positive.
Stator flux (dq): The stator flux linkages \(\varphi_\mathrm{d}\) and \(\varphi_\mathrm{q}\) in the rotating reference frame.
Rotational speed: The rotational speed \(\omega_\mathrm{m}\) of the rotor in radians per second \((\frac{\mathrm{rad}}{\mathrm{s}})\).
Rotor position: The mechanical rotor angle \(\theta_\mathrm{m}\) in radians.
Electrical torque: The electrical torque \(T_\mathrm{e}\) of the machine in newton-meters \((\mathrm{Nm})\).

References

H. Bühler, “Réglage de systèmes d’électronique de puissance, vol 1: Théorie”, Presses Polytechniques et universitaires romandes, Lausanne, 1997.
H. Bühler, “Réglage de systèmes d’électronique de puissance, vol 2: Entraînements réglés”, Presses Polytechniques et universitaires romandes, Lausanne, 1997.