Synchronous reluctance machine with sinusoidal flux distribution

**Library:**Simscape / Electrical / Electromechanical / Reluctance & Stepper

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.

The diagram shows the motor construction with a single pole-pair on the rotor. For
the axes convention shown, when rotor mechanical angle
*θ _{r}* is zero, the

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:

*v*,_{a}*v*, and_{b}*v*are the individual phase voltages across the stator windings._{c}*R*is the equivalent resistance of each stator winding._{s}*i*,_{a}*i*, and_{b}*i*are the currents flowing in the stator windings._{c}*ψ*,_{a}*ψ*, and_{b}*ψ*are the magnetic fluxes that link each stator winding._{c}

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:

*L*,_{aa}*L*, and_{bb}*L*are the self-inductances of the stator windings._{cc}*L*,_{ab}*L*,_{ac}*L*,_{ba}*L*,_{bc}*L*, and_{ca}*L*are the mutual inductances of the stator windings._{cb}

${\theta}_{e}=N{\theta}_{r},$

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

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

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

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

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

and

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

where:

*θ*is the rotor mechanical angle._{r}*θ*is the rotor electrical angle._{e}*L*is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings._{s}*L*is the stator inductance fluctuation. This value is the amplitude of the fluctuation in self-inductance and mutual inductance with changing rotor angle._{m}*M*is the stator mutual inductance. This value is the average mutual inductance between the stator windings._{s}

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 *θ _{e}* 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.*θ*is the rotor mechanical angle._{r}

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:

*i*,_{d}*i*, and_{q}*i*are the_{0}*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

*i*,_{a}*i*, and_{b}*i*are the stator currents._{c}*v*,_{d}*v*, and_{q}*v*are the_{0}*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

*v*,_{a}*v*, and_{b}*v*are the stator currents._{c}The

*dq0*inductances are defined, respectively as${L}_{d}={L}_{s}+{M}_{s}+\frac{3}{2}{L}_{m}$

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

${L}_{0}={L}_{s}-2{M}_{s}$.

*R*is the stator resistance per phase._{s}*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**).*T*is the load torque._{L}*B*is the rotor damping._{m}*ω*is the rotor mechanical rotational speed.*J*is the rotor inertia.

The flux distribution is sinusoidal.

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 (Simscape).

[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.