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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}

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

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

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

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

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

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

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}*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.*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

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

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 block has four optional thermal ports, one for each of the three stator
windings and one for the rotor. These ports are hidden by default. To expose the
thermal ports, right-click the block in your model, select
**Simscape** > **Block
choices**, and then select the desired block variant with thermal
ports: **Composite three-phase ports | Show thermal port** or
**Expanded three-phase ports | Show thermal port**. This
action displays the thermal ports on the block icon, and exposes the
**Thermal** parameters. These parameters are described further
on this reference page.

Use the thermal ports to simulate the effects of copper resistance and iron losses that convert electrical power to heat. For more information on using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

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.

The flux distribution is sinusoidal.

[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.