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:

$[\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}] = [\begin{matrix} R_{s} & 0 & 0 \\ 0 & R_{s} & 0 \\ 0 & 0 & R_{s} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} \frac{d ψ_{a}}{d t} \\ \frac{d ψ_{b}}{d t} \\ \frac{d ψ_{c}}{d t} \end{matrix}],$

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.
$\frac{d ψ_{a}}{d t},$ $\frac{d ψ_{b}}{d t},$ and $\frac{d ψ_{c}}{d t}$ 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:

$[\begin{matrix} ψ_{a} \\ ψ_{b} \\ ψ_{c} \end{matrix}] = [\begin{matrix} L_{a a} & L_{a b} & L_{a c} \\ L_{b a} & L_{b b} & L_{b c} \\ L_{c a} & L_{c b} & L_{c c} \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}] + [\begin{matrix} ψ_{a m} \\ ψ_{b m} \\ ψ_{c m} \end{matrix}],$

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:

$θ_{e} = N_{p} x + r o t o r o f f s e t,$

$L_{a a} = L_{s} + L_{m} cos (2 θ_{e})$

$L_{b b} = L_{s} + L_{m} cos (2 (θ_{e} - 2 π / 3))$

$L_{c c} = L_{s} + L_{m} cos (2 (θ_{e} + 2 π / 3))$

$L_{a b} = L_{b a} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6))$

$L_{b c} = L_{c b} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6 - 2 π / 3))$

and

$L_{c a} = L_{a c} = - M_{s} - L_{m} \cos (2 (θ_{e} + π / 6 + 2 π / 3))$

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.
$N_{p} = \frac{π}{τ}$ 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:

$[\begin{matrix} ψ_{a m} \\ ψ_{b m} \\ ψ_{c m} \end{matrix}] = [\begin{matrix} ψ_{m} \cos θ_{e} \\ ψ_{m} \cos (θ_{e} - 2 π / 3) \\ ψ_{m} \cos (θ_{e} + 2 π / 3) \end{matrix}]$

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:

$P = 2 / 3 [\begin{matrix} \cos θ_{e} & \cos (θ_{e} - 2 π / 3) & \cos (θ_{e} + 2 π / 3) \\ - \sin θ_{e} & - \sin (θ_{e} - 2 π / 3) & - \sin (θ_{e} + 2 π / 3) \\ 0.5 & 0.5 & 0.5 \end{matrix}]$

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:

$[\begin{matrix} v_{d} \\ v_{q} \\ v_{0} \end{matrix}] = P [\begin{matrix} v_{a} \\ v_{b} \\ v_{c} \end{matrix}]$

and

$[\begin{matrix} i_{d} \\ i_{q} \\ i_{0} \end{matrix}] = P [\begin{matrix} i_{a} \\ i_{b} \\ i_{c} \end{matrix}]$

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

$v_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} - N_{p} v i_{q} L_{q},$

$v_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} + N_{p} v (i_{d} L_{d} + ψ_{m}),$

$v_{0} = R_{s} i_{0} + L_{0} \frac{d i_{0}}{d t}$

and

$F = \frac{3}{2} N_{p} (i_{q} (i_{d} L_{d} + ψ_{m}) - i_{d} i_{q} L_{q}),$

$M \frac{d v}{d t} = F - F_{L} - B_{m} v,$

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:

$k_{e} = k_{f} = N_{p} ψ_{m} .$

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

expand all

~ — 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.