Three-phase switched reluctance machine

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

The Switched Reluctance Machine block represents a three-phase switched reluctance machine (SRM). The stator has three pole pairs, carrying the three motor windings, and the rotor has several nonmagnetic poles. The motor produces torque by energizing a stator pole pair, inducing a force on the closest rotor poles and pulling them toward alignment. The diagram shows the motor construction.

Choose this machine in your application to take advantage of these properties:

Low cost

Relatively safe failing currents

Robustness to high temperature operation

High torque-to-inertia ratio

Use this block to model an SRM using easily measurable or estimable parameters. To model an SRM using FEM data, see Switched Reluctance Motor Parameterized with FEM Data.

The rotor stroke angle for a three-phase machine is

${\theta}_{st}=\frac{2\pi}{3{N}_{r}},$

where:

*θ*is the stoke angle._{st}*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,
*λ*(*θ _{ph}*) curve.
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 give this equation, which defines the total instantaneous torque for a three-phase SRM:

$T={\displaystyle \sum}_{j=1}^{3}{T}_{ph}(j).$

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

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

The flux linkage curve is approximated based on parametric and geometric data:

$${\lambda}_{ph}\left({i}_{ph},{\theta}_{ph}\right)={\lambda}_{sat}\left(1-{e}^{-{L}_{0}(\theta ){i}_{ph}/{\lambda}_{sat}}\right),$$

where *L _{0}* is the
unsaturated inductance.

The effects of saturation are more prominent as the product of current and
unsaturated inductance approach the saturated flux linkage value. Specify this
value using the **Saturated flux linkage** parameter.

Differentiating the flux equation then gives the winding inductance:

$$L({\theta}_{ph})={L}_{0}({\theta}_{ph}){e}^{\left(-{L}_{0}({\theta}_{ph}){i}_{ph}/{\lambda}_{sat}\right)}$$

The unsaturated inductance varies between a minimum and maximum value. The minimum value occurs when a rotor pole is directly between two stator poles. The maximum occurs when the rotor pole is aligned with a stator pole. In between these two points, the block approximates the unsaturated inductance linearly as a function of rotor angle. This figure shows the unsaturated inductance as a rotor pole passes over a stator pole.

In the figure:

*θ*corresponds to the angle subtended by the rotor pole. Set it using the_{R}**Angle subtended by each rotor pole**parameter.*θ*corresponds to the angle subtended by the stator pole. Set it using the_{S}**Angle subtended by each stator pole**parameter.*θ*corresponds to the angle subtended by this full cycle, determined by_{C}*2π/2n*where*n*is the number of stator pole pairs.

The block provides four modeling variants. To select the desired variant,
right-click the block in your model. From the context menu, select **Simscape** > **Block choices**, and then one of these variants:

`Composite three-phase ports | No thermal port`

— The block contains composite three-phase electrical conserving ports associated with the stator windings, but does not contain thermal ports. This variant is the default.`Expanded three-phase ports | No thermal port`

— The block contains expanded electrical conserving ports associated with the stator windings, but does not contain thermal ports.`Composite three-phase ports | Show thermal port`

— The block contains composite three-phase electrical conserving ports associated with the stator windings and four thermal conserving ports, one for each of the three windings and one for the rotor.`Expanded three-phase ports | Show thermal port`

— The block contains expanded electrical conserving ports associated with the stator windings and four thermal conserving ports, one for each of the three windings and one for the rotor.

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.

Selecting a thermal block variant exposes thermal parameters.

In practice, magnetic edge effects prevent the inductance from taking a
trapezoidal shape as a rotor pole passes over a stator pole. To model these effects,
and to avoid gradient discontinuities that hinder solver convergence, the block
smooths the gradient *∂L _{0}/∂θ* at inflection
points. To change the angle over which this smoothing is applied, use the

The block assumes that a zero rotor angle corresponds to a rotor pole that is
aligned perfectly with the *a*-phase.

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, Second
Edition*. New York: CRC, 2005.

[2] Ilic'-Spong, M., R. Marino, S. Peresada, and D. Taylor. “Feedback
linearizing control of switched reluctance motors.” *IEEE Transactions
on Automatic Control*. Vol. 32, No. 5, 1987, pp. 371–379.

BLDC | PMSM | Synchronous Machine Field Circuit | Synchronous Machine Measurement | Synchronous Reluctance Machine