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Four or five-phase switched reluctance machine (SRM)

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

The Switched Reluctance Machine (Multi-Phase) block represents a four- or five-phase switched reluctance machine (SRM).

The diagram shows the motor construction for the four-phase machine.

The diagram shows the motor construction for the five-phase machine.

The rotor stroke angle for a multiphase machine is

${\theta}_{st}=\frac{2\pi}{{N}_{s}{N}_{r}}$

where:

*θ*is the stoke angle._{st}*N*is the number of phases._{s}*N*is the number of rotor poles._{r}

The torque production capability, *β*, of one rotor pole is

$\beta =\frac{2\pi}{{N}_{r}}.$

The mathematical model for a switched reluctance machine (SRM) is highly nonlinear due to
influence of the magnetic saturation on the flux linkage-to-angle curve,
*λ*(*θ _{ph}*). The phase
voltage equation for an SRM is

${v}_{ph}={R}_{s}{i}_{ph}+\frac{d{\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right)}{dt}$

where:

*v*is the voltage per phase._{ph}*R*is the stator resistance per phase._{s}*i*is the current per phase._{ph}*λ*is the flux linkage per phase._{ph}*θ*is the angle per phase._{ph}

Rewriting the phase voltage equation in terms of partial derivatives yields this equation:

${v}_{ph}={R}_{s}{i}_{ph}+\frac{\partial {\lambda}_{ph}}{\partial {i}_{ph}}\frac{d{i}_{ph}}{dt}+\frac{\partial {\lambda}_{ph}}{\partial {\theta}_{ph}}\frac{d{\theta}_{ph}}{dt}.$

Transient inductance is defined as

${L}_{t}\left({i}_{ph},{\theta}_{ph}\right)=\frac{\partial {\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right)}{\partial {i}_{ph}},$

or more simply as

$\frac{\partial {\lambda}_{ph}}{\partial {i}_{ph}}.$

Back electromotive force is defined as

${E}_{ph}=\frac{\partial {\lambda}_{ph}}{\partial {\theta}_{ph}}{\omega}_{r}.$

Substituting these terms into the rewritten voltage equation yields this voltage equation:

${v}_{ph}={R}_{s}{i}_{ph}+{L}_{t}\left({i}_{ph},{\theta}_{ph}\right)\frac{d{i}_{ph}}{dt}+{E}_{ph}.$

Applying the co-energy formula to equations for torque,

${T}_{ph}=\frac{\partial W\left({\theta}_{ph}\right)}{\partial {\theta}_{r}},$

and energy,

$W\left({i}_{ph},{\theta}_{ph}\right)=\underset{0}{\overset{{i}_{ph}}{{\displaystyle \int}}}{\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right)d{i}_{ph}$

yields an integral equation that defines the instantaneous torque per phase, that is,

${T}_{ph}\left({i}_{ph},{\theta}_{ph}\right)=\underset{0}{\overset{{i}_{ph}}{{\displaystyle \int}}}\frac{\partial {\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right)}{\partial {\theta}_{ph}}d{i}_{ph}.$

Integrating over the phases gives this equation, which defines the total instantaneous torque as

$T={\displaystyle \sum}_{j=1}^{{N}_{s}}{T}_{ph}(j).$

The equation for motion is

$$J\frac{d\omega}{dt}=T-{T}_{L}-{B}_{m}\omega $$

where:

*J*is the rotor inertia.*ω*is the mechanical rotational speed.*T*is the rotor torque. For the Switched 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}*J*is the rotor inertia.*B*is the rotor damping._{m}

For high-fidelity modeling and control development, use empirical data and finite element calculation to determine the flux linkage curve in terms of current and angle, that is,

${\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right).$

For low-fidelity modeling, you can also approximate the curve using analytical techniques. One such technique [2] uses this exponential function:

${\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right)={\lambda}_{sat}\left(1-{e}^{-{i}_{ph}f({\theta}_{ph})}\right),$

where:

*λ*is the saturated flux linkage._{sat}*f*(*θ*) is obtained by Fourier expansion._{r}

For the Fourier expansion, use the first two even terms of this equation:

$f\left({\theta}_{ph}\right)=a+b\mathrm{cos}\left({N}_{r}{\theta}_{ph}\right)$

where *a* > *b*,

$a=\frac{{L}_{\mathrm{min}}+{L}_{\mathrm{max}}}{2{\lambda}_{sat}},$

and

$b=\frac{{L}_{max}-{L}_{min}}{2{\lambda}_{sat}}.$

A zero rotor angle corresponds to a rotor pole that is aligned perfectly with the
*a*-phase, that is, peak flux.

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] Boldea, I. and S. A. Nasar.
*Electric Drives.* 2nd Ed. New York: CRC Press,
2005.

[2] Iliĉ-Spong, M., R. Marino, S.
Peresada, and D. Taylor. “Feedback linearizing control of switched reluctance
motors.” *IEEE Transactions on Automatic Control*. Vol. 32,
Number 5, 1987, pp. 371–379.