Switched Reluctance Machine

Purpose

Detailed model of switched reluctance machine with open windings

Library

Electrical / Machines

Description

These components represent analytical models of three common switched reluctance machine types: three-phase 6/4 SRM, four-phase 8/6 SRM and five-phase 10/8 SRM.

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. In the component icon, the positive terminals of the stator windings are marked with a dot.

Note

The Switched Reluctance Machine models can only be simulated with the Continuous State-Space Method.

The machine flux linkage is modeled as a non-linear function of the stator current and rotor angle \(\Psi(i, \theta)\) accounting for both the magnetization characteristic of the iron and the variable air gap.

Fig. 235 Magnetization characteristics

In the unaligned rotor position the flux linkage is approximated as a linear function:

\[\Psi_\mathrm{u}(i) = L_\mathrm{u} \cdot i\]

In the aligned rotor position the flux linkage is a non-linear function of the stator current:

\[\Psi_\mathrm{a}(i) = \Psi_\mathrm{sat} \cdot (1 - e^{-K \cdot i}) + L_\mathrm{sat} \cdot i\]

where

\[K = \frac{L_\mathrm{a} - L_\mathrm{sat}}{\Psi_\mathrm{sat}}\]

For intermediate rotor positions the flux linkage is written as a weighted sum of these two extremes

\[\Psi(i,\theta) = \Psi_\mathrm{u}(i) + f(\theta) \cdot (\Psi_\mathrm{a}(i)-\Psi_\mathrm{u}(i))\]

using the weighting function

\[f(\theta) = \frac{1}{2}+\frac{1}{2}\cos\bigg(N_r\bigg[\theta + 2\pi \frac{x}{N_\mathrm{s}}\bigg]\bigg)\]

where \(N_r\) is the number of rotor poles, \(N_s\) is the number of stator poles, and \(x=0\ldots(N_s/2-1)\) is the index of the stator phase.

Electrical System

Fig. 236 SRM equivalent circuit schema

The terminal voltage of a stator phase is determined by the equation

\[v = R \cdot i + \frac{d\Psi}{dt} = R \cdot i + \frac{\partial \Psi}{\partial i} \cdot \frac{di}{dt} + \frac{\partial \Psi}{\partial \theta} \cdot \frac{d\theta}{dt}\]

The electromagnetic torque produced by one phase is the derivative of the coenergy with respect to the rotor angle:

\[T(i,\theta) = \frac{\partial}{\partial \theta}\int_0^i\Psi(i',\theta)di'\]

The total torque \(T_\mathrm{e}\) of the machine is given by the sum of the individual phase torques.

Mechanical System

Rotor speed:

\[\frac{d}{dt}\omega = \frac{1}{J} ( T_\mathrm{e} - F \omega - T_\mathrm{m} )\]

Rotor angle:

\[\frac{d}{dt}\theta = \omega\]

Parameters

Stator resistance

Stator resistance \(R\) in ohms (\(\Omega\)).

Unaligned stator inductance

Stator inductance \(L_\mathrm{u}\) in the unaligned rotor position, in henries (\(\mathrm{H}\)).

Initial aligned stator inductance

Initial stator inductance \(L_\mathrm{a}\) in the aligned rotor position, in henries (\(\mathrm{H}\)).

Saturated aligned stator inductance

Saturated stator inductance \(L_\mathrm{sat}\) in the aligned rotor position, in henries (\(\mathrm{H}\)).

Aligned saturation flux linkage

Flux linkage \(\Phi_\mathrm{sat}\) at which the stator saturates in the aligned position, in (\(\mathrm{Vs}\)).

Inertia

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

Friction coefficient

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

Initial rotor speed

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

Initial rotor angle

Initial mechanical rotor angle \(\theta_\mathrm{m,0}\) in radians.

Initial stator currents

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

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.

Back EMF

The back EMF voltages \(e_\mathrm{a}\), \(e_\mathrm{b}\), \(e_\mathrm{c}\) in volts \((\mathrm{V})\).

Stator flux linkage

The flux linkages in the individual phases of the machine in \((\mathrm{Vs})\).

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

• D.A. Torrey, J.A. Lang, “Modelling a nonlinear variable-reluctance motor drive”, IEE Proceedings, Vol. 137, Pt. B, No. 5, Sept. 1990.

• D.A. Torrey, X.-M. Niu, E.J. Unkauf, “Analytical modelling of variable-reluctance machine magnetisation characteristic”, IEE Proceedings Electric Power Applications, Vol. 142, No. 1, Jan. 1995.