Engee documentation

PMSM

Permanent magnet synchronous motor (PMSM) with sinusoidal shape of back EMF.

pmsm

Description

The PMSM block simulates a synchronous motor with permanent magnets on the rotor and a three-phase stator with star or delta windings with sinusoidal back EMF. The figure shows the equivalent circuit diagram for the stator windings:

permanent magnet 1

You can also model a PMSM in a delta or star configuration by setting the Winding type parameter to Delta-wound or Wye-wound respectively.

Motor design

The permanent magnets of the rotor produce a magnetic field that varies in a sinusoidal pattern.

permanent magnet 2

When the Rotor angle definition is set to Angle between the a-phase magnetic axis and the d-axis', the fluxes of phase A and the permanent magnets are equalised at zero rotor mechanical angle θ . When the Rotor angle definition parameter is set to `Angle between the a-phase magnetic axis and the q-axis, the rotor mechanical angle is equal to the angle between the A-phase magnetic axis and the q-axis of the rotor.

Equations

Voltages on the stator windings:

ψψψ ,

where:

  • , , - stator winding voltage;

  • - equivalent resistance of each stator winding;

  • , , - currents in the winding;

  • ψ , ψ , ψ - stator winding flux differentials.

The stator flux is generated by two sources: flux from the rotor permanent magnets and currents in the stator inductance. Thus:

ψψψψψψ ,

where:

  • ψψψ - stator winding flux-coupling of the stator windings;

  • , , - stator phase inductances;

  • , , etc. - mutual inductances between the stator phases;

  • ψ , ψ , ψ - flux-couplings of the permanent magnet with the corresponding stator phases.

The stator winding inductances are functions of the electric angle of the rotor and are determined as follows:

θθθ

θ

θ

θ

θ

θ

θ

Where:

  • θ - mechanical angle of the rotor;

  • θ - electric angle of the rotor;

  • θ - is equal to 0 if the electric angle of the rotor is set with respect to the axis , or -π/2 if the electric angle of the rotor is set with respect to the axis ;

  • - stator winding inductance. This value is the average value of the inductance of each of the stator windings;

  • - stator inductance fluctuation. The oscillation of the stator winding eigeninductances ( , , ) and the oscillation of the mutual inductances ( , , ) between each two stator windings depend on the change of the rotor angle and always have the same value, i.e. ;

  • - mutual inductance of the stator windings. The value is the average value of the mutual inductance between the stator windings ( , , ).

The flux-coupling of the permanent magnet of the phase winding А is maximum at θ and zero at θ . Thus, the motor flux-coupling is determined by:

ψψψψθψθψθ ,

where ψ is the flux-coupling of the permanent magnet.

Simplified electrical equations

Applying the Park-Gorev transformation to the block of electrical equations allows us to obtain an expression for the torque independent of the rotor rotation angle.

The Park-Gorev transformation is defined as follows:

θθπθπθθπθπ ,

where θ is the electric angle, defined as θ . - is the number of pole pairs.

Applying the Park-Gorev transformation to the voltages and currents of the stator windings leads them to the coordinate system dq0, which is independent of the rotor angle:

Applying the Park-Goreff transformation to the first two electrical equations allows us to proceed to the other equations that determine the behaviour of the block:

ω

ωψ

ψ ,

Where:

  • - inductance of the d axis of the stator;

  • - stator q axis inductance;

  • - stator zero-sequence inductance;

  • ω - mechanical speed of rotor rotation;

  • - number of rotor pole pairs;

  • - rotor torque. The torque is transferred from the motor housing (block port ) to the motor rotor (block port ).

The PMSM block uses an original, non-orthogonal implementation of the Park-Gorev transformation.

Alternative parametrization of the flux-current coil

In addition to the direct method of setting the flux ratio of the permanent magnet, there are other parameterization options.

The motor parameters can be set with back EMF or torque, which are most often specified in the motor data sheets, using the parameter Permanent magnet flux linkage.

The Permanent back EMF is the peak voltage produced by the permanent magnet per unit speed of each phase. The relationship between the peak permanent magnet flux linkage and back EMF is as follows:

ψ .

The back EMF, , for one phase is:

ω .

The Torque constant is the peak torque induced by the current unit of each phase. It is numerically identical to the back EMF constant if both quantities are expressed in SI units:

ψ .

When and the currents in all three phases are balanced, the total rotor torque T is equal to:

,

where is the peak current in any of the three windings.

The factor follows from the fact that in steady state it is the sum of the torques from all phases. Therefore, the torque constant can also be:

,

where T is the measured total torque when tested with balanced three-phase current with peak line voltage . The rms value of the line current is:

.

Ports

~ - three-phase port
electricity

Expandable three-phase port.

n - neutral
electricity

An electrical port associated with the neutral.

Dependencies

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

R is the motor rotor
`Rotational Mechanics

The mechanical port of rotation associated with the motor rotor.

C - motor housing
`rotational mechanics

A mechanical port of rotation associated with the motor housing.

Parameters

Main

Winding type - winding configuration
Wye-wound (by default) | Delta-wound

Select the winding configuration:

  • Wye-wound - star connection.

  • Delta-wound - delta connection. Phase is connected between ports a and b, phase is connected between ports b and c, phase is connected between ports c and a.

Modelling fidelity is a way of specifying machine parameters
Constant Ld, Lq and PM (by default).

Select the machine parameter setting method:

  • Constant Ld, Lq and PM - the values , and are constant and are determined by the respective parameters.

Number of pole pairs - number of pole pairs
6 (By default)

Number of pole pairs of the rotor.

Permanent magnet flux linkage parameterization - permanent magnet flux linkage parameterization
| Specify flux linkage (by default) | Specify torque constant | Specify back EMF constant.

Select Specify flux linkage, Specify torque constant or Specify back EMF constant.

Permanent magnet flux linkage - permanent magnet flux linkage
0.03 Vb (by default).

Peak value of permanent magnet flux linkage with either stator winding.

Dependencies

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

Torque constant, N*m/A - torque constant
0.18 N*m/A (by default).

Torque constant for either stator winding.

Dependencies

To enable the parameter set Permanent magnet flux linkage parameterization to Specify torque linkage and Modelling fidelity to Constant Ld, Lq and PM.

Back EMF constant, V*s/rad - back EMF constant
0.18 V*s/rad (by default).

Back EMF constant for either stator winding.

The back EMF is the peak voltage of each phase produced by the permanent magnet per unit speed of rotation.

Dependencies

To enable the 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 (by default) | Specify Ls, Lm, and Ms.

Method of stator parameterization.

Dependencies

Select Specify Ld, Lq and L0 or Specify Ls, Lm and Ms.

Stator d-axis inductance Ld, H - stator d axis inductance
0.00022 Gn (by default).

The inductance along the d axis.

Dependencies

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

Stator q-axis inductance Lq, H is the inductance along the q axis of the stator
0.00022 Gn (by default).

The inductance along the q axis.

Dependencies

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

Stator zero-sequence inductance L0, H - stator zero-sequence inductance
0.00016 Gn (by default)

Zero-sequence inductance.

Dependencies

To use this parameter, set:

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

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

Stator self-inductance per phase Ls, H - stator self-inductance per phase
0.0002 Gn (by default).

Stator self-inductance per phase.

Dependencies

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

Stator inductance fluctuation Lm, H - stator inductance fluctuation
-0.0002 Gn (By default).

It is taken into account that self-inductance and mutual inductance of stator windings change with change of rotor angle.

Dependencies

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

SStator mutual inductance Ms, H - stator mutual inductance
0.00002 Gn (By default).

Average mutual inductance between stator windings.

Dependencies

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

Stator resistance per phase Rs, Ohm - stator phase resistance
0.013 Ohm (by default).

Stator phase resistance.

Dependencies

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

Zero sequence - zero sequence option
Include (by default) | Exclude.

Option to include or exclude zero sequence.

  • Include - includes zero-sequence terms. To prioritise model accuracy, use this setting by default. Using this option:

    • Causes an error when modelling using the Partitioning solver.

    • Expands the zero sequence parameter in the Impedances settings.

  • Exclude - excludes zero sequence conditions. To improve simulation speed for desktop simulation or real-time deployment, select this option.

Dependencies

This option is used when the Winding Type parameter is set to Wye-wound.

Rotor angle definition - reference point for rotor angle measurement
Angle between phase A magnetic axis and d-axis (by default) | Angle between phase A magnetic axis and q-axis.

Reference point for measuring the rotor angle.

When Angle between the a-phase magnetic axis and the d-axis is selected, the rotor and phase A fluxes coincide when the rotor rotation angle is zero.

When `Angle between the a-phase magnetic axis and the q-axis' is selected, the A phase current produces the maximum torque when the rotor rotation angle is zero.

Mechanical

Rotor inertia, kg*m^2 - rotor inertia
0.01 kg*m^2 (by default) | `non-negative scalar `

Inertia of the rotor attached to the mechanical port R. The value can be zero.

RRotor damping, N*m*s/rad - rotor damping
0 (by default) | non-negative scalar.

Rotational damping.

Initial values

D-axis current - d-axis current
0 (By default)

Initial value of current along -axis .

Q-axis current - q-axis current
`0 (By default)

Initial value of current along axis .

Torque - torque .
`0 (By default)

Initial torque.

Rotor speed - rotor speed
0 (By default)

Initial rotor speed.

Rotor angle - rotor rotation angle
0 (By default)

Initial rotor angle.

Examples