Switched Reluctance Machine (Multi-Phase)

Four, five, or six-phase switched reluctance machine (SRM)

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Libraries:
Simscape / Electrical / Electromechanical / Reluctance & Stepper

Description

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

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

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

This diagram shows the motor construction for the six-phase machine.

Equations

The rotor stroke angle for a multiphase machine is

${\theta }_{st}=\frac{2\pi }{N_s{N}_r}$

where:

• θ_st is the stoke angle.

• N_s is the number of phases.

• 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 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_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 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_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

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

Calculating Iron Losses

The Switched Reluctance Machine (Multi-Phase) block implements the iron losses as a reduction in the electrical torque.

If you set the Stator parameterization parameter to Specify parametric data, specify the Magnetizing resistance parameter to model iron losses.

If you set the Stator parameterization parameter to Specify tabulated flux data, the block tabulates the losses as the power lost in each stator phase. The losses for each stator only depends on the stator current in that phase, the rotor angle (geometrically shifted to that phase), and the rotor speed. The iron loss power of the rotor is computed as a percentage of the total iron loss power, as specified in the Percentage of total iron losses associated with the rotor parameter.

If you tabulate the losses with current, speed, and angle, the iron loss torque reduces the electrical torque of a generic "A" phase:

${\tau }_{elecA}=\tau \left(i_A,{\theta }_A\right)-{\tau }_{loss}\left(i_A,w,{\theta }_A\right)$

where ${\tau }_{loss}\left(i_A,w,{\theta }_A\right)=sign\left(w\right)\frac{P_{loss}\left(abs\left(i_A\right),abs\left(w\right),{\theta }_A\right)}{abs\left(w\right)+w_{min}}$ is the torque loss computed as the mechanical power loss divided by the rotor speed. w_min is the minimum rotor speed and it is equal to 1 rad/s. This value prevents a division by zero when the rotor speed is equal to 0.

If you tabulate the losses in function of only current and speed, the iron loss torque reduces the electrical torque of a generic "A" phase:

${\tau }_{loss}\left(i_A,w\right)=sign\left(w\right)\frac{P_{loss}\left(abs\left(i_A\right),abs\left(w\right)\right)}{abs\left(w\right)+w_{min}}.$

The iron losses are applied to the mechanical side as a reduction in torque. If you create the iron loss table, P_loss, from FEM results, the FEM tool might express the power lost as electrical power instead of mechanical power.

The total electrical torque is equal to the sum of the electrical torque of each phase:

${\tau }_{elec}={\tau }_{elecA}+{\tau }_{elecB}+{\tau }_{elecC}.$

The total power dissipated is equal to the sum of the iron losses and the copper losses:

$P_{dissipated}=\left({\tau }_{lossA}+{\tau }_{lossB}+{\tau }_{lossC}\right)w+R_a{i}_a^2+R_b{i}_b^2+R_c{i}_c^2.$

Iron losses with thermal ports

If you expose the thermal ports of the block, the power dissipated in each phase act as heat sources in the motor and in the rotor:

$\begin{array}{l}{P}_{dissA}=R_a{i}_a^2+P_{ironA}\\ P_{dissB}=R_b{i}_b^2+P_{ironB}\\ P_{dissC}=R_c{i}_c^2+P_{ironC}\\ P_{dissR}=P_{ironR}\end{array}$

The iron loss in the rotor is equal to a percentage of the total iron losses. This loss remains constant and does not depend on other states:

$P_{ironR}=\frac{percen{t}_{rotor}}{100}\left({\tau }_{lossA}+{\tau }_{lossB}+{\tau }_{lossC}\right)w.$

Then the remaining iron loss is distributed among the different stator phases:

$$\begin{array}{l}{P}_{ironA}=\left(1-\frac{percen{t}_{rotor}}{100}\right){\tau }_{lossA}w\\ P_{ironB}=\left(1-\frac{percen{t}_{rotor}}{100}\right){\tau }_{lossB}w\\ P_{ironC}=\left(1-\frac{percen{t}_{rotor}}{100}\right){\tau }_{lossC}w\end{array}$$

Model Thermal Effects

You can expose thermal ports to model the effects of losses that convert power to heat. To expose the thermal ports, set the Modeling option parameter to either:

• No thermal port — The block does not contain thermal ports.

• Show thermal port — The block contains multiple thermal conserving ports.

For more information about using thermal ports in actuator blocks, see Simulating Thermal Effects in Rotational and Translational Actuators.

Assumptions

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

Variables

To set the priority and initial target values for the block variables before simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. You can specify nominal values using different sources, including the Nominal Values section in the block dialog box or Property Inspector. For more information, see System Scaling by Nominal Values.

Examples

Four-Phase Switched Reluctance Machine Control

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

• Open Model

Five-Phase Switched Reluctance Machine Control

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

• Open Model

Six-Phase Switched Reluctance Machine Control

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

• Open Model

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.

• Open Model

Ports

Conserving

R — Machine rotor

mechanical rotational

Mechanical rotational conserving port associated with the machine rotor.

Data Types: double

C — Machine case

mechanical rotational

Mechanical rotational conserving port associated with the machine case.

Data Types: double

a1 — a-phase positive supply

electrical

Electrical positive supply for phase-a.

Data Types: double

b1 — b-phase positive supply

electrical

Electrical positive supply for phase-b.

Data Types: double

c1 — c-phase positive supply

electrical

Electrical positive supply for phase-c.

Data Types: double

d1 — d-phase positive supply

electrical

Electrical positive supply for phase-d.

Data Types: double

e1 — e-phase positive supply

electrical

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

f1 — f-phase positive supply

electrical

Electrical positive supply for phase-f.

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

Data Types: double

a2 — a-phase negative supply

electrical

Electrical negative supply for phase-a.

Data Types: double

b2 — b-phase negative supply

electrical

Electrical negative supply for phase-b.

Data Types: double

c2 — c-phase negative supply

electrical

Electrical negative supply for phase-c.

Data Types: double

d2 — d-phase negative supply

electrical

Electrical negative supply for phase-d.

Data Types: double

e2 — e-phase negative supply

electrical

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

f2 — f-phase negative supply

electrical

Electrical negative supply for phase-f.

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

Data Types: double

HA — a-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding a.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HB — b-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding b.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HC — c-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding c.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HD — d-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding d.

Dependencies

To enable this port, set Modeling option to Show thermal port.

HE — e-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding e.

Dependencies

To enable this port, set Modeling option to Show thermal port and Number of stator phases to Five-phase.

HF — f-phase terminal thermal port

thermal

Thermal conserving port associated with stator winding f.

Dependencies

To enable this port, set Modeling option to Show thermal port and Number of stator phases to Six-phase.

HR — Rotor thermal port

thermal

Thermal conserving port associated with the rotor.

Dependencies

To enable this port, set Modeling option to Show thermal port.

Parameters

Modeling option — Whether to enable thermal ports

No thermal port (default) | Show thermal port

Whether to enable the thermal ports of the block and model the effects of losses that convert power to heat.

Main

Number of stator phases — Four- or five-phase SRM

Four-phase (default) | Five-phase | Six-phase

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

Number of rotor poles — Rotor pole

8 (default) | positive integer

Number of pole pairs on the rotor.

Stator resistance per phase — Resistance

3 Ohm (default) | positive scalar

Per-phase resistance of each of the stator windings.

Stator parameterization — Parameterization method

Specify saturated flux linkage (default) | Specify flux characteristic

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 — Flux linkage

0.43 Wb (default) | positive scalar

Saturated flux linkage per phase.

Dependencies

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

Aligned inductance — Inductance

0.0046 H (default) | positive scalar

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.

Unaligned inductance — Inductance

6.7e-4 H (default) | positive scalar

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, i — Current

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

Current vector used to identify the flux linkage curve family.

Dependencies

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

Angle vector, theta — Angle

[0, 22.5, 45] deg (default) | positive vector

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, Phi(i,theta) — Flux

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

Flux linkage matrix that defines the flux linkage curve family.

Dependencies

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

Iron Losses

For more information on calculating iron losses, see Calculating Iron Losses.

Model — Iron losses modeling option

None (default) | Tabulate losses from current and speed | Tabulate losses from current, speed, and angle

Whether to enable iron losses modeling.

Dependencies

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

Magnetizing resistance — Magnetizing resistance for iron losses

inf Ohm (default)

Magnetizing resistance. The value must be greater than zero. The default value is inf, which implies that there are no iron losses.

Dependencies

To enable this parameter, set Stator parameterization to Specify parametric data.

Rotor speed vector, w — Rotor speed

[500, 1000, 1500] rpm (default)

Vector of rotor speed at which the iron loss is tabulated.

Dependencies

To enable this parameter, set Model to either Tabulate losses from current and speed or Tabulate losses from current, speed, and angle.

Iron loss table, Ploss(i,w) — Iron loss table tabulated on current and speed

zeros(3, 3) W (default)

Mechanical power lost in the shaft due to iron losses in a particular phase, tabulated as a function of the instantaneous phase current and the rotor speed.

Dependencies

To enable this parameter, set Model to Tabulate losses from current and speed.

Iron loss table, Ploss(i,w,theta) — Iron loss table tabulated on current, speed, and angle

zeros(3, 3, 2) W (default)

Mechanical power lost in the shaft due to iron losses in a particular phase, tabulated as a function of the instantaneous phase current, the rotor speed, and the rotor angle.

Dependencies

To enable this parameter, set Model to Tabulate losses from current, speed, and angle.

Mechanical

Rotor inertia — Inertia

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

Inertia of the rotor attached to mechanical translational port R.

Rotor Damping — Damping

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

Rotary damping.

Thermal

To enable these parameters, set Modeling option to Show thermal port.

Measurement temperature — Rated temperature

298.15 k (default) | real scalar

Temperature for which motor parameters are quoted.

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.

Thermal mass for each stator winding — Winding thermal mass

100 J/K (default) | positive scalar

Thermal mass value for each stator winding. The thermal mass is the energy required to raise the temperature by one degree.

Rotor thermal mass — Rotor thermal mass

200 J/K (default) | positive scalar

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

Percentage of total iron losses associated with the rotor — Percentage of total iron losses

50 (default)

Percentage of total iron losses associated with the rotor.

Dependencies

To enable this parameter, set Model to either Tabulate losses from current and speed or Tabulate losses from current, speed, and angle.

References

[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.

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Version History

Introduced in R2018b