# Switched Reluctance Machine

Switched reluctance machine (SRM)

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## Description

The Switched Reluctance Machine block represents a three-phase switched reluctance machine (SRM). The diagram shows the motor construction.

### Equations

The rotor stroke angle for a three-phase machine is

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

where:

• θst is the stoke angle.

• 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, λ(θph) 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:

• 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 give this equation, which defines the total instantaneous torque for a three-phase SRM:
`$T=\sum _{j=1}^{3}{T}_{ph}\left(j\right).$`

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 [2] 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

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

## 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`

Data Types: `double`

Data Types: `double`

## Parameters

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### Main

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:

• Aligned inductance

• Unaligned inductance

Selecting `Specify flux characteristic` enables these parameters:

• Current vector, i

• Angle vector, theta

#### Dependencies

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

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```.

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```.

#### Dependencies

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

### Mechanical

Moment of inertia of the rotor.

Damping of the rotor.

### Initial Conditions

Initial a-, b-, and c-phase currents.

The initial angle of the rotor.

Initial speed of the rotor. If the rotor inertia, J, is zero, the initial speed of the rotor is zero rpm and the initial rotor speed is ignored.

## References

[1] Boldea, I. and S. A. Nasar. Electric Drives, Second Edition. CRC Press, New York, 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.