Main Content

Permanent magnet linear synchronous motor with sinusoidal flux distribution

**Library:**Simscape / Electrical / Electromechanical / Permanent Magnet

The PMLSM block models a permanent magnet linear synchronous motor with a three-phase wye-wound stator. The 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`

.

This figure shows the motor construction.

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*, and_{b}*v*are the individual phase voltages across the stator windings._{c}*R*is the equivalent resistance of each stator winding._{s}*i*,_{a}*i*, and_{b}*i*are the currents flowing in the stator windings._{c}$$\frac{d{\psi}_{a}}{dt},$$$$\frac{d{\psi}_{b}}{dt},$$ 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*, and_{b}*ψ*are the total fluxes linking each stator winding._{c}*L*,_{aa}*L*, and_{bb}*L*are the self-inductances of the stator windings._{cc}*L*,_{ab}*L*,_{ac}*L*,_{ba}*L*,_{bc}*L*, and_{ca}*L*are the mutual inductances of the stator windings._{cb}*ψ*,_{am}*ψ*, and_{bm}*ψ*are the permanent magnet fluxes linking the stator windings._{cm}

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

${\theta}_{e}={N}_{p}x+rotor\text{\hspace{0.17em}}offset,$

${L}_{aa}={L}_{s}+{L}_{m}\text{cos}(2{\theta}_{e}),$

${L}_{bb}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{e}-2\pi /3\right)),$

${L}_{cc}={L}_{s}+{L}_{m}\text{cos}(2\left({\theta}_{e}+2\pi /3\right)),$

$${L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6\right)\right),$$

${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6-2\pi /3\right)\right),$

and

${L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2\left({\theta}_{e}+\pi /6+2\pi /3\right)\right),$

where:

*θ*is the electrical angle._{e}*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*is the per-phase stator self-inductance. This value is the average self-inductance of each of the stator windings._{s}*L*is the stator inductance fluctuation. This value is the fluctuation in self-inductance and mutual inductance with changing angle._{m}*M*is the stator mutual inductance. This value is the average mutual inductance between the stator windings._{s}

The permanent magnet flux linking winding *a* is a maximum when
*θ _{e}* = 0° and zero when

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

where *ψ _{m}* is the permanent magnet flux
linkage.

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:

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

where *θ _{e}* is the electrical angle defined
as

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*+ 3/2_{s}*L*._{m}*L*is the stator_{d}*d*-axis inductance.*L*=_{q}*L*+_{s}*M*− 3/2_{s}*L*._{m}*L*is the stator_{q}*q*-axis inductance.*L*=_{0}*L*– 2_{s}*M*._{s}*L*is the stator zero-sequence inductance._{0}*R*is the stator resistance per phase._{s}*v*is the linear speed.*N*is the polar pitch factor._{p}*M*is the mass of the mover.*B*is the damping._{m}*F*is the load force._{L}

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

${k}_{e}={k}_{f}={N}_{p}{\psi}_{m}.$

The block has four optional thermal ports, one for each of the three stator
windings and one for the rotor. These ports are hidden by default. To expose the
thermal ports, right-click the block in your model, select
**Simscape** > **Block
choices**, and then select **Show thermal
port**. This action displays the thermal ports on the block icon,
and exposes the **Thermal** parameters. These parameters are
described further on this reference page.

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.

[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.