# Synchronous Reluctance Machine

Synchronous reluctance machine with sinusoidal flux distribution

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## Description

The Synchronous Reluctance Machine block represents a synchronous reluctance machine (SynRM) with sinusoidal flux distribution. The figure shows the equivalent electrical circuit for the stator windings.

### Motor Construction

The diagram shows the motor construction with a single pole-pair on the rotor. For the axes convention shown, when rotor mechanical angle θr is zero, the a-phase and permanent magnet fluxes are aligned. The block supports a second rotor axis definition for which rotor mechanical angle is defined as the angle between the a-phase magnetic axis and the rotor q-axis.

### Equations

The combined voltage across the stator windings is

`$\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{ccc}{R}_{s}& 0& 0\\ 0& {R}_{s}& 0\\ 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi }_{a}}{dt}\\ \frac{d{\psi }_{b}}{dt}\\ \frac{d{\psi }_{c}}{dt}\end{array}\right],$`
where:

• va, vb, and vc are the individual phase voltages across the stator windings.

• Rs is the equivalent resistance of each stator winding.

• ia, ib, and ic are the currents flowing in the stator windings.

• ψa, ψb, and ψc are the magnetic fluxes that link each stator winding.

The permanent magnet, excitation winding, and the three stator windings contribute to the flux that links each winding. The total flux is defined as

`$\left[\begin{array}{c}{\psi }_{a}\\ {\psi }_{b}\\ {\psi }_{c}\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& {L}_{ab}& {L}_{ac}\\ {L}_{ba}& {L}_{bb}& {L}_{bc}\\ {L}_{ca}& {L}_{cb}& {L}_{cc}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]$`
where:

• Laa, Lbb, and Lcc are the self-inductances of the stator windings.

• Lab, Lac, Lba, Lbc, Lca, and Lcb are the mutual inductances of the stator windings.

The inductances in the stator windings are functions of rotor electrical angle and are defined as

`${L}_{aa}={L}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{r}\right),$`
`${L}_{bb}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{r}-\frac{2\pi }{3}\right)\right),$`
`${L}_{cc}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{r}+\frac{2\pi }{3}\right)\right),$`
`${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left({\theta }_{r}+\frac{\pi }{6}\right),$`
`${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left({\theta }_{r}+\frac{\pi }{6}-\frac{2\pi }{3}\right),$`
`${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left({\theta }_{r}+\frac{\pi }{6}+\frac{2\pi }{3}\right),$`
where:

• Ls is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings.

• Lm is the stator inductance fluctuation. This value is the amplitude of the fluctuation in self-inductance and mutual inductance with changing rotor angle.

• θr is the rotor mechanical angle.

• Ms is the stator mutual inductance. This value is the average mutual inductance between the stator windings.

### Simplified Equations

Applying the Park transformation to the block electrical defining equations produces an expression for torque that is independent of rotor angle.

The Park transformation, K, is defined as

`$K=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]$`
where θe is the electrical angle. The electrical angle depends on the rotor mechanical angle and the number of pole pairs such that
`${\theta }_{e}=N{\theta }_{r},$`
where:

• N is the number of pole pairs.

• θr is the rotor mechanical angle.

The inverse of the Park transformation is defined by

`${K}^{-1}=\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& -\mathrm{sin}{\theta }_{e}& 1\\ \mathrm{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& 1\\ \mathrm{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}+\frac{2\pi }{3}\right)& 1\end{array}\right].$`

Applying the Park transformation to the first two electrical defining equations produces equations that define the behavior of the block:

`${v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}-N\omega {i}_{q}{L}_{q},$`
`${v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega {i}_{d}{L}_{d},$`
`${v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt},$`
`$T=\frac{3}{2}N\left({i}_{q}{i}_{d}{L}_{d}-{i}_{d}{i}_{q}{L}_{q}\right)$`
`$J\frac{d\omega }{dt}=T-{T}_{L}-{B}_{m}\omega ,$`
where:

• id, iq, and i0 are the d-axis, q-axis, and zero-sequence currents, defined by

`$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]=P\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right],$`
where ia, ib, and ic are the stator currents.

• vd, vq, and v0 are the d-axis, q-axis, and zero-sequence currents, defined by

`$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]=P\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right],$`
where va, vb, and vc are the stator currents.

• The dq0 inductances are defined, respectively as

• ${L}_{d}={L}_{s}+{M}_{s}+\frac{3}{2}{L}_{m}$

• ${L}_{q}={L}_{s}+{M}_{s}-\frac{3}{2}{L}_{m}$

• ${L}_{0}={L}_{s}-2{M}_{s}$.

• Rs is the stator resistance per phase.

• N is the number of rotor pole pairs.

• T is the rotor torque. For the Synchronous Reluctance Machine block, torque flows from the machine case (block conserving port C) to the machine rotor (block conserving port R).

• TL is the load torque.

• Bm is the rotor damping.

• ω is the rotor mechanical rotational speed.

• J is the rotor inertia.

### Assumptions

The block assumes that the flux distribution is sinusoidal.

## Ports

### Conserving

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Mechanical rotational conserving port associated with the machine rotor.

Mechanical rotational conserving port associated with the machine case.

Expandable three-phase port associated with the stator windings.

Electrical conserving port associated with the neutral phase.

## Parameters

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### Main

Number of permanent magnet pole pairs on the rotor.

Method for parameterizing the stator.

#### Dependencies

Selecting `Specify Ld, Lq and L0` enables these parameters:

• Stator d-axis inductance, Ld

• Stator q-axis inductance, Lq

• Stator zero-sequence inductance, L0

Selecting `Specify Ls, Lm, and Ms` enables these parameters:

• Stator self-inductance per phase, Ls

• Stator inductance fluctuation, Lm

• Stator mutual inductance, Ms

Direct-axis inductance of the machine stator.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq and L0```.

Quadrature-axis inductance of the machine stator.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq and L0```.

Zero-axis inductance for the machine stator.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq and L0```.

Average self-inductance of the three stator windings.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ls, Lm, and Ms```.

Amplitude of the fluctuation in self-inductance and mutual inductance with the rotor angle.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ls, Lm, and Ms```.

Average mutual inductance between the stator windings.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ls, Lm, and Ms```.

Resistance of each of the stator windings.

Resistance of the field winding.

### Mechanical

Inertia of the rotor.

Damping of the rotor.

### Initial Conditions

The a-, b-, and c-phase currents at simulation start time.

Rotor angle at simulation start time.

Initial speed of the rotor. If the rotor inertia, J, is zero, the initial speed of the rotor is zero rpm and the initial rotor speed is ignored.

## References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Mbayed, R. Analysis of Faulted Power Systems. Hoboken, NJ: Wiley-IEEE Press, 1995.

[3] Anderson, P. M. Contribution to the Control of the Hybrid Excitation Synchronous Machine for Embedded Applications. Universite de Cergy Pontoise, Neuville sur Oise, France, 2012.