# Synchronous Reluctance Machine

Synchronous reluctance machine with sinusoidal flux distribution

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• Simscape / Electrical / Electromechanical / Reluctance & Stepper

## 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.

${\theta }_{e}=N{\theta }_{r}+rotor\text{\hspace{0.17em}}offset$

${L}_{aa}={L}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}\right),$

${L}_{bb}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-2\pi /3\right)\right),$

${L}_{cc}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+2\pi /3\right)\right),$

${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{e}+\pi /6\right)\right),$

${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{e}+\pi /6-2\pi /3\right)\right),$

and

${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta }_{e}+\pi /6+2\pi /3\right)\right),$

where:

• θr is the rotor mechanical angle.

• θe is the rotor electrical angle.

• rotor offset is 0 if you define the rotor electrical angle with respect to the d-axis, or -pi/2 if you define the rotor electrical angle with respect to the q-axis.

• 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.

• 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, P, is defined as

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

where θe is the electrical angle. The electrical angle depends on the rotor mechanical angle and the number of pole pairs such that

${\theta }_{e}=N{\theta }_{r}+rotor\text{\hspace{0.17em}}offset$

where:

• N is the number of pole pairs.

• θr is the rotor mechanical angle.

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 flux distribution is sinusoidal.

### Variables

Use the Variables settings to specify the priority and initial target values for the block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

## 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.

Stator parameterization model.

#### Dependencies

The Stator parameterization setting affects the visibility of other parameters.

Select the modeling fidelity:

• Constant Ld and LqLd and Lq values are constant and defined by their respective parameters.

• Tabulated Ld and LqLd and Lq values are computed online from DQ currents look-up tables as follows:

${L}_{d}={f}_{1}\left({i}_{d},{i}_{q}\right)$

${L}_{d}={f}_{2}\left({i}_{d},{i}_{q}\right)$

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Direct-axis inductance of the machine stator.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Constant Ld and Lq.

Quadrature-axis inductance of the machine stator.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Constant Ld and Lq.

Direct-axis current vector, iD.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Ld matrix.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Lq matrix.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0 and the Modeling fidelity parameter to Tabulated Ld and Lq.

Zero-axis inductance for the machine stator.

#### Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ld, Lq and L0 and Zero sequence is set to Include.

Average self-inductance of the three stator windings.

#### Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ls, Lm, and Ms.

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

#### Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ls, Lm, and Ms.

Average mutual inductance between the stator windings.

#### Dependencies

This parameter is visible only if Stator parameterization is set to Specify Ls, Lm, and Ms.

Resistance of each of the stator windings.

Zero-sequence model:

• Include — Prioritize model fidelity. An error occurs if you Include zero-sequence terms for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.

• Exclude — Prioritize simulation speed for desktop simulation or real-time deployment.

#### Dependencies

If this parameter is set to:

• Include and Stator parameterization is set to Specify Ld, Lq, and L0 — The Stator zero-sequence inductance, L0 parameter is visible.

• Exclude — The Stator zero-sequence inductance, L0 parameter is not visible.

### Mechanical

Inertia of the rotor attached to mechanical translational port R.

Rotary damping.

Reference point for the rotor angle measurement. If you select the default value, the rotor and a-phase fluxes are aligned for a zero-rotor angle. Otherwise, an a-phase current generates the maximum torque value for a zero-rotor angle.

## References

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

[2] Anderson, P. M. Analysis of Faulted Power Systems. Hoboken, NJ: Wiley-IEEE Press, 1995.

[3] Moghaddam, R. Synchronous Reluctance Machine (SynRM) in Variable Speed Drives (VSD) Applications - Theoretical and Experimental Reevaluation. KTH School of Electrical Engineering, Stockholm, Sweden, 2011.