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Switched Reluctance Machine (Multi-Phase)

Four or five-phase switched reluctance machine (SRM)

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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. Equations

The rotor stroke angle for a multiphase machine is

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

where:

• θst is the stoke angle.

• Ns is the number of phases.

• Nr is the number of rotor poles.

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(θph). 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:

• vph is the voltage per phase.

• Rs is the stator resistance per phase.

• iph is the current per phase.

• λph is the flux linkage per phase.

• θph is the angle per phase.

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,

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

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

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

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

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).

• TL is the load torque.

• J is the rotor inertia.

• Bm is the rotor damping.

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  uses this exponential function:

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

where:

• λsat is the saturated flux linkage.

• f(θr) is obtained by Fourier expansion.

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,

and

Assumptions

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

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 (Simscape).

Ports

Conserving

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Mechanical rotational conserving port associated with the machine rotor.

Data Types: double

Mechanical rotational conserving port associated with the machine case.

Data Types: double

Electrical positive supply for phase-a.

Data Types: double

Electrical positive supply for phase-b.

Data Types: double

Electrical positive supply for phase-c.

Data Types: double

Electrical positive supply for phase-d.

Data Types: double

Electrical positive supply for phase-e.

This port is visible if you select Five-phase for the Number of stator phases parameter.

Data Types: double

Electrical negative supply for phase-a.

Data Types: double

Electrical negative supply for phase-b.

Data Types: double

Electrical negative supply for phase-c.

Data Types: double

Electrical negative supply for phase-d.

Data Types: double

Electrical negative supply for phase-e.

This port is visible if you select Five-phase for the Number of stator phases parameter.

Data Types: double

Parameters

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Main

Type of multiphase SRM in terms of the number of stator phases.

Dependencies

Selecting Five-phase enables these ports:

• e1

• e2

Number of pole pairs on the rotor.

Per-phase resistance of each of the stator windings.

Method for parameterizing the stator.

Dependencies

Selecting Specify saturated flux linkage enables these parameters:

• Saturated flux linkage

• Aligned inductance

• Unaligned inductance

Selecting Specify flux characteristic enables these parameters:

• Current vector, i

• Angle vector, theta

• Flux linkage matrix, Phi(i,theta)

Saturated flux linkage per phase.

Dependencies

To enable this parameter, set Stator parameterization to Specify saturated flux linkage.

Inductance when the axis of the rotor pole is identical to the axis of the excited stator pole. The value of this parameter must be greater than the value of the Unaligned inductance parameter.

Dependencies

To enable this parameter, set Stator parameterization to Specify saturated flux linkage.

Inductance when the axis between two rotor poles is identical to the axis of the excited stator pole. The value of this parameter must be less than the value of the Aligned inductance parameter.

Dependencies

To enable this parameter, set Stator parameterization to Specify saturated flux linkage.

Current vector used to identify the flux linkage curve family.

Dependencies

To enable this parameter, set Stator parameterization to Specify flux characteristic.

Angle vector used to identify the flux linkage curve family.

Dependencies

To enable this parameter, set Stator parameterization to Specify flux characteristic.

Flux linkage matrix that defines the flux linkage curve family.

Dependencies

To enable this parameter, set Stator parameterization to Specify flux characteristic.

Mechanical

Inertia of the rotor attached to mechanical translational port R.

Rotary damping.

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

 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.