Switched Reluctance Machine

Three-phase switched reluctance machine

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Description

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.

Equations

Switched Reluctance Machine Block

The rotor stroke angle for a three-phase machine is

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

where:

• θ_st is the stoke angle.

• N_r 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(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_ph is the voltage per phase.

• R_s is the stator resistance per phase.

• i_ph is the current per phase.

• λ_ph is the flux linkage per phase.

• θ_ph 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,

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

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_L is the load torque.

• J is the rotor inertia.

• B_m 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:

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

where:

• λ_sat is the saturated flux linkage.

• f(θ_r) is obtained by Fourier expansion.

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

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

where a > b,

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

and

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

Switched Reluctance Motor Block

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 L₀ 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_R corresponds to the angle subtended by the rotor pole. Set it using the Angle subtended by each rotor pole parameter.

• θ_S corresponds to the angle subtended by the stator pole. Set it using the Angle subtended by each stator pole parameter.

• θ_C corresponds to the angle subtended by this full cycle, determined by 2π/2n where n is the number of stator pole pairs.

Modeling Variants

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.

Dependencies

Selecting a thermal block variant exposes thermal parameters.

Numerical Smoothing

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 ∂L₀/∂θ at inflection points. To change the angle over which this smoothing is applied, use the Angle over which flux gradient changes are smoothed parameter.

Assumptions

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

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.

Ports

Conserving

~1 — Positive three-phase composite terminal

electrical

Electrical conserving three-phase port associated with the positive terminals of the stator windings.

Dependencies

This port is exposed if you select one of these model variants:

• Composite three-phase ports | No thermal port

• Composite three-phase ports | Show thermal port

~2 — Negative three-phase composite terminal

electrical

Electrical conserving three-phase port associated with the negative terminals of the stator windings.

Dependencies

This port is exposed if you select one of these model variants:

• Composite three-phase ports | No thermal port

• Composite three-phase ports | Show thermal port

a1 — Positive a-phase terminal

electrical

Electrical conserving port associated with the positive terminal of stator winding a.

Dependencies

This port is exposed if you select one of these model variants:

• Expanded three-phase ports | No thermal port

• Expanded three-phase ports | Show thermal port

a2 — Negative a-phase terminal

electrical

Electrical conserving port associated with the negative terminal of stator winding a.

Dependencies

This port is exposed if you select one of these model variants:

• Expanded three-phase ports | No thermal port

• Expanded three-phase ports | Show thermal port

b1 — Positive b-phase terminal

electrical

Electrical conserving port associated with the positive terminal of stator winding b.

Dependencies

This port is exposed if you select one of these model variants:

• Expanded three-phase ports | No thermal port

• Expanded three-phase ports | Show thermal port

b2 — Negative b-phase terminal

electrical

Electrical conserving port associated with the negative terminal of stator winding b.

Dependencies

This port is exposed if you select one of these model variants:

• Expanded three-phase ports | No thermal port

• Expanded three-phase ports | Show thermal port

c1 — Positive c-phase terminal

electrical

Electrical conserving port associated with the positive terminal of stator winding c.

Dependencies

This port is exposed if you select one of these model variants:

• Expanded three-phase ports | No thermal port

• Expanded three-phase ports | Show thermal port

c2 — Negative c-phase terminal

electrical

Electrical conserving port associated with the negative terminal of stator winding c.

Dependencies

This port is exposed if you select one of these model variants:

• Expanded three-phase ports | No thermal port

• Expanded three-phase ports | Show thermal port

R — Rotor

mechanical

Mechanical rotational conserving port associated with the rotor.

C — Casing

mechanical

Mechanical rotational conserving port associated with the stator or casing.

HA — a-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding a.

Dependencies

This port is exposed if you select one of these model variants:

• Composite three-phase ports | Show thermal port

• Expanded three-phase ports | Show thermal port

HB — b-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding b.

Dependencies

This port is exposed if you select one of these model variants:

• Composite three-phase ports | Show thermal port

• Expanded three-phase ports | Show thermal port

HC — c-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding c.

Dependencies

This port is exposed if you select one of these model variants:

• Composite three-phase ports | Show thermal port

• Expanded three-phase ports | Show thermal port

HR — Rotor thermal port

thermal

Thermal conserving port associated with the rotor.

Dependencies

This port is exposed if you select one of these model variants:

• Composite three-phase ports | Show thermal port

• Expanded three-phase ports | Show thermal port

Parameters

Main

Number of rotor poles — Rotor poles

4 (default) | positive integer

Number of rotor poles.

Stator resistance per phase — Resistance

3 Ohm (default) | positive scalar

Per-phase resistance of each of the stator windings.

Stator parameterization — Parameterization method

Specify parametric data (default) | Specify parametric and geometric data | Specify tabulated flux data

Method for parameterizing the stator.

Dependencies

Selecting Specify parametric data enables these parameters:

• Magnetizing resistance

• Saturated flux linkage

• Aligned inductance

• Unaligned inductance

Selecting Specify parametric and geometric data enables these parameters:

• Magnetizing resistance

• Saturated flux linkage

• Aligned inductance

• Unaligned inductance

• Angle subtended by each stator pole

• Angle subtended by each rotor pole

• Angle over which flux gradient changes are smoothed

Selecting Specify tabulated flux data enables these parameters:

• Current vector, i

• Angle vector, theta

• Flux linkage matrix, Phi(i,theta)

Magnetizing resistance — Magnetic losses

inf Ohm (default) | real scalar

The total magnetizing resistance for each of the phase windings. The default value inf indicates that there are no iron losses.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric data or Specify parametric and geometric data.

Saturated flux linkage — Flux linkage

0.43 Wb (default) | positive scalar

Saturated flux linkage per phase.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric data or Specify parametric and geometric data.

Aligned inductance — Inductance

0.0046 H (default) | positive scalar

The value of this parameter must be greater than the value of the Unaligned inductance parameter.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric data or Specify parametric and geometric data.

Unaligned inductance — Inductance

6.7e-4 H (default) | positive scalar

The value of this parameter must be less than the value of the Aligned inductance parameter.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric data or Specify parametric and geometric data.

Angle subtended by each stator pole — Stator tooth angle

45 deg (default) | positive scalar

Angle spanned by each stator tooth. This value must be greater than or equal to the value of Angle subtended by each rotor pole.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric and geometric data.

Angle subtended by each rotor pole — Rotor tooth angle

42 deg (default) | positive scalar

Angle spanned by each rotor tooth.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric and geometric data.

Angle over which flux gradient changes are smoothed — Inflection point smoothing

0.5 deg (default) | positive scalar

Angle over which sharp edges in trapezoidal inductance curve are smoothed. This value must be smaller than the value of Angle subtended by each rotor pole.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify parametric and geometric data.

Current vector, i — Current

[0, 50, 100] A (default) | vector

Current vector used to identify the flux linkage curve family.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify tabulated flux data.

Angle vector, theta — Angle

[0, 45, 90] deg (default) | vector

Angle vector used to identify the flux linkage curve family.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify tabulated flux data.

Flux linkage matrix, Phi(i,theta) — Flux

[0, 0, 0; .37, .06, .37; .43, .1, .43] Wb (default) | matrix

Flux linkage matrix that defines the flux linkage curve family.

Dependencies

This parameter is exposed when you set Stator parameterization to Specify tabulated flux data.

Mechanical

Rotor inertia — Inertia

0.01 kg*m^2 (default) | zero or positive scalar

Inertia of the rotor attached to mechanical translational port R.

Rotor Damping — Damping

0 N*m/(rad/s) (default) | positive scalar

Rotary damping.

Thermal

These parameters appear only for blocks with exposed thermal ports.

Resistance temperature coefficient — Temperature coefficient

3.93e-3 1/K (default) | positive scalar

Coefficient α in the equation relating resistance to temperature for all three windings, as described in Thermal Model for Actuator Blocks. The default value, 3.93e-3 1/K, is for copper.

Measurement temperature — Rated temperature

25 degC (default) | real scalar

The temperature for which motor parameters are quoted.

Winding thermal mass — Winding thermal mass

100 J/K (default) | positive scalar

The thermal mass value for the a-, b-, and c-windings. The thermal mass is the energy required to raise the temperature by one degree.

Rotor thermal mass — Rotor thermal mass

50 J/K (default) | positive scalar

The thermal mass of the rotor. The thermal mass is the energy required to raise the temperature of the rotor by one degree.

Percentage of magnetizing resistance associated with the rotor — Iron loss heating distribution

50 (default) | positive scalar

The percentage of the magnetizing resistance associated with the magnetic path through the rotor. This parameter determines how much of the iron loss heating is attributed to:

• The rotor thermal port HR

• The three stator thermal ports HA, HB, and HC

Model Examples

Switched Reluctance Machine Speed Control

Control the rotor speed in a switched reluctance machine (SRM) based electrical drive. A DC voltage source feeds the SRM through a controlled three-arm bridge. The converter turn-on and turn-off angles are maintained constant.

Switched Reluctance Motor Parameterized with FEM Data

Build a model of a Switched Reluctance Motor (SRM). A custom Simscape™ composite component is used to construct the three stator pole pairs using the Simscape Electrical™ FEM-Parameterized Rotary Actuator as a base component. The states of the input pins (reverse, on/off) control the torque applied to the motor shaft.

References

[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.

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