PMLSM

Permanent magnet linear synchronous motor with sinusoidal flux distribution

Since R2020a

• 

Libraries:
Simscape / Electrical / Electromechanical / Permanent Magnet

Description

The PMLSM block models a permanent magnet linear synchronous motor (PMLSM) with a three-phase wye-wound stator. Use this block to model linear synchronous motors (LSMs) and linear servo motors. This figure shows the equivalent electrical circuit for the stator windings.

You can also model the permanent magnet linear synchronous motor in a delta-wound configuration by setting Winding type to Delta-wound.

Motor Construction

This figure shows the motor construction.

Equations

Voltages across the stator windings are defined by:

$$\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]=\left[\begin{array}{ccc}{R}_s& 0& 0\\ 0& R_s& 0\\ 0& 0& R_s\end{array}\right]\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi }_a}{dt}\\ \frac{d{\psi }_b}{dt}\\ \frac{d{\psi }_c}{dt}\end{array}\right],$$

where:

• v_a, v_b, and v_c are the individual phase voltages across the stator windings.

• R_s is the equivalent resistance of each stator winding.

• i_a, i_b, and i_c are the currents flowing in the stator windings.

• $$\frac{d{\psi }_a}{dt},$$$$\frac{d{\psi }_b}{dt},$$ and $$\frac{d{\psi }_c}{dt}$$ are the rates of change of magnetic flux in each stator winding.

The permanent magnet and the three windings contribute to the total flux linking each winding. The total flux is defined by:

$$\left[\begin{array}{c}{\psi }_a\\ {\psi }_b\\ {\psi }_c\end{array}\right]=\left[\begin{array}{ccc}{L}_{aa}& L_{ab}& L_{ac}\\ L_{ba}& L_{bb}& L_{bc}\\ L_{ca}& L_{cb}& L_{cc}\end{array}\right]\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{c}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\end{array}\right],$$

where:

• ψ_a, ψ_b, and ψ_c are the total fluxes linking each stator winding.

• L_aa, L_bb, and L_cc are the self-inductances of the stator windings.

• L_ab, L_ac, L_ba, L_bc, L_ca, and L_cb are the mutual inductances of the stator windings.

• ψ_am, ψ_bm, and ψ_cm are the permanent magnet fluxes linking the stator windings.

The inductances in the stator windings are functions of electrical angle, and are defined by:

${\theta }_e=N_{p}x+rotoroffset,$

$L_{aa}=L_s+L_m\text{cos}(2{\theta }_e)$

$L_{bb}=L_s+L_m\text{cos}(2\left({\theta }_e-2\pi /3\right))$

$L_{cc}=L_s+L_m\text{cos}(2\left({\theta }_e+2\pi /3\right))$

$$L_{ab}=L_{ba}=-M_s-L_m\cos \left(2\left({\theta }_e+\pi /6\right)\right)$$

$L_{bc}=L_{cb}=-M_s-L_m\cos \left(2\left({\theta }_e+\pi /6-2\pi /3\right)\right)$

and

$L_{ca}=L_{ac}=-M_s-L_m\cos \left(2\left({\theta }_e+\pi /6+2\pi /3\right)\right)$

where:

• θ_e is the electrical angle.

• rotor offset is 0 if you define the rotor electrical angle with respect to the d-axis, or -pi/2 if you define the rotor electrical angle with respect to the q-axis.

• $$N_p=\frac{\pi }{\tau }$$ and τ is the polar pitch.

• x is the distance.

• L_s is the per-phase stator self-inductance. This value is the average self-inductance of each of the stator windings.

• L_m is the stator inductance fluctuation. This value is the fluctuation in self-inductance and mutual inductance with changing angle.

• M_s is the stator mutual inductance. This value is the average mutual inductance between the stator windings.

The permanent magnet flux linking winding a is a maximum when θ_e = 0° and zero when θ_e = 90°. Therefore, the linked motor flux is defined by:

$$\left[\begin{array}{c}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\end{array}\right]=\left[\begin{array}{c}{\psi }_m\cos {\theta }_e\\ {\psi }_m\cos \left({\theta }_e-2\pi /3\right)\\ {\psi }_m\cos \left({\theta }_e+2\pi /3\right)\end{array}\right]$$

where ψ_m is the permanent magnet flux linkage.

Simplified Electrical Equations

Applying Park’s transformation to the electrical equations produces an expression for force that is independent of the angle.

Park’s transformation is defined by:

$P=2/3\left[\begin{array}{ccc}\cos {\theta }_e& \cos \left({\theta }_e-2\pi /3\right)& \cos \left({\theta }_e+2\pi /3\right)\\ -\sin {\theta }_e& -\sin \left({\theta }_e-2\pi /3\right)& -\sin \left({\theta }_e+2\pi /3\right)\\ 0.5& 0.5& 0.5\end{array}\right]$

where θ_e is the electrical angle defined as N_px.

Using Park's transformation on the stator winding voltages and currents transforms them into the dq0 frame, which is independent of the angle:

$$\left[\begin{array}{c}{v}_d\\ v_q\\ v_0\end{array}\right]=P\left[\begin{array}{c}{v}_a\\ v_b\\ v_c\end{array}\right]$$

and

$\left[\begin{array}{c}{i}_d\\ i_q\\ i_0\end{array}\right]=P\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]$

Applying Park’s transformation to the first two electrical equations produces the following equations that define the block behavior:

$v_d=R_s{i}_d+L_d\frac{d{i}_d}{dt}-N_{p}v{i}_q{L}_q,$

$v_q=R_s{i}_q+L_q\frac{d{i}_q}{dt}+N_{p}v(i_d{L}_d+{\psi }_m),$

$v_0=R_s{i}_0+L_0\frac{d{i}_0}{dt}$

and

$F=\frac{3}{2}{N}_p\left(i_q\left(i_d{L}_d+{\psi }_m\right)-i_d{i}_q{L}_q\right),$

$M\frac{dv}{dt}=F-F_L-B_{m}v,$

where:

• L_d = L_s + M_s + 3/2 L_m. L_d is the stator d-axis inductance.

• L_q = L_s + M_s − 3/2 L_m. L_q is the stator q-axis inductance.

• L₀ = L_s – 2M_s. L₀ is the stator zero-sequence inductance.

• R_s is the stator resistance per phase.

• v is the linear speed.

• N_p is the polar pitch factor.

• M is the mass of the mover.

• B_m is the damping.

• F_L is the load force.

The PMLSM block uses the original, nonorthogonal implementation of the Park transform. If you try to apply the alternative implementation, you get different results for the dq0 voltage and currents.

The relationship between the force constant k_f, the back-emf constant k_e, and the permanent magnet flux linkage is defined as follows:

$k_e=k_f=N_p{\psi }_m.$

Model Thermal Effects

You can expose thermal ports to model the effects of generated heat and motor temperature. To expose the thermal ports, set the Modeling option parameter to either:

• No thermal port — The block contains expanded electrical conserving ports associated with the stator windings, but does not contain thermal ports.

• Show thermal port — The block contains expanded electrical conserving ports associated with the stator windings and thermal conserving ports for each of the windings and for the rotor.

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

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

Three-Phase PMLSM Drive

Control the position in a three-phase permanent magnet linear synchronous machine (PMLSM) drive. The Control subsystem uses a PI-based cascade control structure with an outer position control loop, a speed control loop, and two inner current control loops. A controlled three-phase converter feeds the PMLSM. The simulation uses step references. The Scopes subsystem contains scopes that allow you to see the simulation results.

• Open Model

Ports

Conserving

~ — Three-phase port

electrical

Expandable three-phase port.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Wye-wound or Delta-wound.

a — a-phase

electrical

Electrical conserving port associated with a-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

b — b-phase

electrical

Electrical conserving port associated with b-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

c — c-phase

electrical

Electrical conserving port associated with c-phase.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Wye-wound or Delta-wound.

~1 — Stator positive-end connections

electrical

Since R2024a

Expandable three-phase port associated with the stator positive-end connections.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Open-end.

~2 — Stator negative-end connections

electrical

Since R2024a

Expandable three-phase port associated with the stator negative-end connections.

Dependencies

To enable this port, set Electrical connection to Composite three-phase ports and Winding type to Open-end.

a1 — a positive-end connection

electrical

Since R2024a

Electrical conserving port associated with the a positive-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

b1 — b positive-end connection

electrical

Since R2024a

Electrical conserving port associated with the b positive-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

c1 — c positive-end connection

electrical

Since R2024a

Electrical conserving port associated with the c positive-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

a2 — a negative-end connection

electrical

Since R2024a

Electrical conserving port associated with the a negative-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

b2 — b negative-end connection

electrical

Since R2024a

Electrical conserving port associated with the b negative-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

c2 — c negative-end connection

electrical

Since R2024a

Electrical conserving port associated with the c negative-end connection.

Dependencies

To enable this port, set Electrical connection to Expanded three-phase ports and Winding type to Open-end.

n — Neutral phase

electrical

Electrical conserving port associated with the neutral phase.

Dependencies

To enable this port, set Winding Type to Wye-wound and Zero sequence to Include.

R — Motor mover

mechanical

Mechanical translational conserving port associated with the motor mover.

C — Motor case

mechanical

Mechanical translational conserving port associated with the motor case.

HA — Winding A thermal port

thermal

Thermal conserving port associated with winding A.

Dependencies

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

HB — Winding B thermal port

thermal

Thermal conserving port associated with winding B.

Dependencies

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

HC — Winding C thermal port

thermal

Thermal conserving port associated with winding C.

Dependencies

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

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 generated heat and motor temperature.

Main

Electrical connection — Electrical connection

Composite three-phase ports (default) | Expanded three-phase ports

Whether to have composite or expanded three-phase ports.

Winding type — Windings configuration

Wye-wound (default) | Delta-wound | Open-end

Configuration for the windings:

• Wye-wound — The windings are wye-wound.

• Delta-wound — The windings are delta-wound. The a-phase is connected between ports a and b, the b-phase between ports b and c and the c-phase between ports c and a.

• Open-end — The windings are in an open-end configuration. The block exposes each end of the windings as electrical connections.

Modeling fidelity — Modeling fidelity

Constant Ld, Lq and PM (default) | Tabulated Ld, Lq and PM

Modeling fidelity:

• Constant Ld, Lq and PM — The Ld, Lq, and PM values are constant and defined by their respective parameters.

• Tabulated Ld, Lq and PM — The Ld, Lq, and PM values are computed online from DQ current lookup tables as follows:h

  $$L_d=f_1(i_d,i_q)$$

  $$L_d=f_2(i_d,i_q)$$

  $${\lambda }_{PM}=f_2(i_d,i_q)$$

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0.

Polar pitch — Polar pitch

0.03 m (default) | scalar

Polar pitch.

Permanent magnet flux linkage parameterization — Permanent magnet flux linkage parameterization

Specify flux linkage (default) | Specify force constant | Specify back EMF constant

Permanent magnet flux linkage, specified as Specify flux linkage, Specify force constant, or Specify back EMF constant.

Permanent magnet flux linkage — Permanent magnet flux linkage

0.03 Wb (default) | scalar

Peak permanent magnet flux linkage with any of the stator windings.

Dependencies

To enable this parameter, set Permanent magnet flux linkage parameterization to Specify flux linkage and Modeling fidelity to Constant Ld, Lq and PM.

Force constant — Force constant

0.18 N/A (default) | scalar

Force constant for any of the stator windings.

Dependencies

To enable this parameter, set Permanent magnet flux linkage parameterization to Specify force constant and Modeling fidelity to Constant Ld, Lq and PM.

Back EMF constant — Back EMF constant

0.18 s*V/m (default) | scalar

Back EMF constant for any of the stator windings.

Dependencies

To enable this parameter, set Permanent magnet flux linkage parameterization to Specify back EMF constant and Modeling fidelity to Constant Ld, Lq and PM.

Stator parameterization — Stator parameterization

Specify Ld, Lq, and L0 (default) | Specify Ls, Lm, and Ms

Stator parameterization, specified as Specify Ld, Lq, and L0 or Specify Ls, Lm, and Ms.

Stator d-axis inductance, Ld — Stator d-axis inductance

0.00019 H (default) | scalar

d-axis inductance.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld, Lq and PM.

Stator q-axis inductance, Lq — Stator q-axis inductance

0.00025 H (default) | scalar

q-axis inductance.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Constant Ld, Lq and PM.

Stator zero-sequence inductance, L0 — Stator zero-sequence inductance

0.00016 H (default) | scalar

Zero-sequence inductance.

Dependencies

To enable this parameter either:

• Set Winding Type to Wye-wound, Zero sequence to Include, and Stator parameterization to Specify Ld, Lq, and L0.

• Set Winding Type to Delta-wound and Stator parameterization to Specify Ld, Lq, and L0.

Direct-axis current vector, iD — Direct-axis current vector

[-200, 0, 200] A (default) | vector

Direct-axis current vector, iD.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Quadrature-axis current vector, iQ — Quadrature-axis current vector

[-200, 0, 200] A (default) | vector

Quadrature-axis current vector, iQ.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Ld matrix, Ld(id,iq) — Ld matrix

0.00019 * ones(3, 3) H (default) | matrix

Ld matrix.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Lq matrix, Lq(id,iq) — Lq matrix

0.00025 * ones(3, 3) H (default) | matrix

Lq matrix.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ld, Lq, and L0 and Modeling fidelity to Tabulated Ld, Lq and PM.

Permanent magnet flux linkage, PM(id,iq) — Permanent magnet flux linkage

0.03 * ones(3, 3) Wb (default)

Permanent magnet flux linkage.

Dependencies

To enable this parameter, set:

• Stator parameterization to Specify Ld, Lq, and L0

• Modeling fidelity to Tabulated Ld, Lq and PM

• Permanent magnet flux linkage parameterization to Specify flux linkage

Force constant matrix, kf(iD,iQ) — Force constant matrix

0.18 * ones(3, 3) N/A (default) | matrix

Force constant matrix.

Dependencies

To enable this parameter, set:

• Stator parameterization to Specify Ld, Lq, and L0

• Modeling fidelity to Tabulated Ld, Lq and PM

• Permanent magnet flux linkage parameterization to Specify force constant

Back EMF constant matrix, ke(iD,iQ) — Back EMF constant matrix

0.18 * ones(3, 3) s*V/m (default) | matrix

Back EMF constant matrix.

Dependencies

To enable this parameter, set:

• Stator parameterization to Specify Ld, Lq, and L0

• Modeling fidelity to Tabulated Ld, Lq and PM

• Permanent magnet flux linkage parameterization to Specify back EMF constant

Stator self-inductance per phase, Ls — Stator self-inductance per phase

0.0002 H (default) | scalar

Average self-inductance of each of the five stator windings.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

Stator inductance fluctuation, Lm — Stator inductance fluctuation

-0.00002 H (default) | scalar

Fluctuation in self-inductance and mutual inductance of the stator windings with angle.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

Stator mutual inductance, Ms — Stator mutual inductance

0.00002 H (default) | scalar

Average mutual inductance between the stator windings.

Dependencies

To enable this parameter, set Stator parameterization to Specify Ls, Lm, and Ms.

Stator resistance per phase, Rs — Stator resistance per phase

0.013 Ohm (default) | scalar

Resistance of each of the stator windings.

Zero sequence — Zero sequence option

Include (default) | Exclude

Option to include or exclude zero-sequence terms.

• Include — Include zero-sequence terms. To prioritize model fidelity, use this default setting. Using this option:

  • Results in an error for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.

  • Exposes a zero-sequence parameter.

• Exclude — Exclude zero-sequence terms. To prioritize simulation speed for desktop simulation or real-time deployment, select this option.

Dependencies

To enable this parameter, set Winding Type to Wye-wound.

Angle definition — Reference point for angle measurement

Angle between the a-phase magnetic axis and the d-axis (default) | Angle between the a-phase magnetic axis and the q-axis

Reference point for the angle measurement. The default value is Angle between the a-phase magnetic axis and the d-axis. This definition is shown in the Motor Construction figure. When you select this value, the mover and a-phase fluxes are aligned when the mover angle is zero.

The other value you can choose for this parameter is Angle between the a-phase magnetic axis and the q-axis. When you select this value, the a-phase current generates maximum force when the mover angle is zero.

Mechanical

Mass of the mover — Mass of the mover

0.01 kg (default)

Mass of the motor mover.

Linear damping — Linear damping

0 N/(m/s) (default)

Linear damping.

Thermal

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

Measurement temperature — Measurement temperature

298.15 K (default)

The temperature for which motor parameters are quoted.

Resistance temperature coefficient — Resistance temperature coefficient

3.93e-3 1/K (default)

Coefficient α in the equation relating resistance to temperature, as described in Thermal Model for Actuator Blocks. The default value is for copper.

Permanent magnet flux temperature coefficient — Permanent magnet flux temperature coefficient

-0.001 1/K (default)

Fractional rate of change of permanent magnet flux density with temperature. Use this parameter to linearly reduce the torque and the induced back EMF as temperature rises.

Thermal mass for each stator winding — Thermal mass for each stator winding

100 J/K (default)

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

200 J/K (default)

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

References

[1] Kundur, P. Power System Stability and Control. New York, NY: McGraw Hill, 1993.

[2] Anderson, P. M. Analysis of Faulted Power Systems. IEEE Press Power Systems Engineering Series, 1995. ISBN 0-7803-1145-0.

[3] X. Zhang and J. Pan. Nonlinear Robust Sliding Mode Control for PM Linear Synchronous Motors. 2006 CES/IEEE 5th International Power Electronics and Motion Control Conference, Shanghai, 2006, pp. 1–5.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2020a

R2024a: Open-end configuration for stator windings

To model the stator windings of the machine in an open-end configuration, set the Winding type parameter to Open-end. The PMLSM block then exposes each end of the windings by enabling the corresponding electrical conserving ports.

R2021b: Electrical connection ports update

From R2021b forward, to switch between composite and expanded ports, set the Electrical connection parameter to either Composite three-phase ports or Expanded three-phase ports.

As a result of these changes, inside a model saved in an earlier release, review the Electrical connection parameter of this block.