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# PMSM

Permanent magnet synchronous motor with sinusoidal flux distribution

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• Simscape / Electrical / Electromechanical / Permanent Magnet

## Description

The PMSM block models a permanent magnet synchronous motor with a three-phase wye-wound stator. The figure shows the equivalent electrical circuit for the stator windings.

### Motor Construction

This figure shows the motor construction with a single pole-pair on the rotor.

Permanent magnets generate a rotor magnetic field that creates a sinusoidal rate of change of flux with rotor angle.

For the axes convention in the preceding figure, the a-phase and permanent magnet fluxes are aligned when rotor mechanical angle, θr, is zero. The block supports a second rotor axis definition in which rotor mechanical angle is defined as the angle between the a-phase magnetic axis and the rotor q-axis.

### 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:

• va, vb, and vc are the individual phase voltages across the stator windings.

• Rs is the equivalent resistance of each stator winding.

• ia, ib, and ic 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:

• ψa, ψb, and ψc are the total fluxes linking each stator winding.

• Laa, Lbb, and Lcc are the self-inductances of the stator windings.

• Lab, Lac, Lba, and so on, are the mutual inductances of the stator windings.

• ψam, ψbm, and ψcm are the permanent magnet fluxes linking the stator windings.

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

${\theta }_{e}=N{\theta }_{r},$

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

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

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

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

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

and

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

where:

• θr is the rotor mechanical angle.

• θe is the rotor electrical angle.

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

• Lm is the stator inductance fluctuation. This value is the amplitude of the fluctuation in self-inductance and mutual inductance with changing rotor angle.

• Ms 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 θe = 0° and zero when θe = 90°. Therefore, the linked motor flux is defined by:

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

where ψm is the permanent magnet flux linkage.

### Simplified Electrical Equations

Applying Park’s transformation to the block electrical equations produces an expression for torque that is independent of the rotor 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 θe is the electrical angle defined as r. N is the number of pole pairs.

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

${v}_{d}={R}_{s}{i}_{d}+{L}_{d}\frac{d{i}_{d}}{dt}-N\omega {i}_{q}{L}_{q},$

${v}_{q}={R}_{s}{i}_{q}+{L}_{q}\frac{d{i}_{q}}{dt}+N\omega \left({i}_{d}{L}_{d}+{\psi }_{m}\right),$

${v}_{0}={R}_{s}{i}_{0}+{L}_{0}\frac{d{i}_{0}}{dt},$

and

$T=\frac{3}{2}N\left({i}_{q}\left({i}_{d}{L}_{d}+{\psi }_{m}\right)-{i}_{d}{i}_{q}{L}_{q}\right),$

where:

• Ld = Ls + Ms + 3/2 Lm. Ld is the stator d-axis inductance.

• Lq = Ls + Ms − 3/2 Lm. Lq is the stator q-axis inductance.

• L0 = Ls – 2Ms. L0 is the stator zero-sequence inductance.

• ω is the rotor mechanical rotational speed.

• N is the number of rotor permanent magnet pole pairs.

• T is the rotor torque. Torque flows from the motor case (block physical port C) to the motor rotor (block physical port R).

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

You can parameterize the motor using the back EMF or torque constants which are more commonly given on motor datasheets by using the Permanent magnet flux linkage option.

The back EMF constant is defined as the peak voltage induced by the permanent magnet in each of the phases per unit rotational speed. It is related to peak permanent magnet flux linkage by:

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

From this definition, it follows that the back EMF eph for one phase is given by:

${e}_{ph}={k}_{e}\omega .$

The torque constant is defined as the peak torque induced by each of the phases per unit current. It is numerically identical in value to the back EMF constant when both are expressed in SI units:

${k}_{t}=N{\psi }_{m}.$

When Ld=Lq, and when the currents in all three phases are balanced, it follows that the combined torque T is given by:

$T=\frac{3}{2}{k}_{t}{i}_{q}=\frac{3}{2}{k}_{t}{I}_{pk},$

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

The factor 3/2 follows from this being the steady-state sum of the torques from all phases. Therefore the torque constant kt could also be defined as:

${k}_{t}=\frac{2}{3}\left(\frac{T}{{I}_{pk}}\right),$

where T is the measured total torque when testing with a balanced three-phase current with peak line voltage Ipk. Writing in terms of RMS line voltage:

${k}_{t}=\sqrt{\frac{2}{3}}\left(\frac{T}{{i}_{line,rms}}\right).$

### Variables

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 (Simscape).

## Ports

### Conserving

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Electrical conserving port associated with the neutral phase.

Mechanical rotational conserving port associated with the motor rotor.

Mechanical rotational conserving port associated with the motor case.

## Parameters

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### Main

Select the 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.

Number of permanent magnet pole pairs on the rotor.

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

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

#### Dependencies

This parameter is visible only when you set the Permanent magnet flux linkage parameter to Specify flux linkage.

Torque constant with any of the stator windings.

#### Dependencies

This parameter is visible only when you set the Permanent magnet flux linkage parameter to Specify torque constant.

Back EMF constant with any of the stator windings.

#### Dependencies

This parameter is visible only when you set the Permanent magnet flux linkage parameter to Specify back EMF constant.

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

D-axis inductance.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Q-axis inductance.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

Zero-sequence inductance.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ld, Lq, and L0.

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

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ls, Lm, and Ms.

Amplitude of the fluctuation in self-inductance and mutual inductance of the stator windings with rotor angle.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ls, Lm, and Ms.

Average mutual inductance between the stator windings.

#### Dependencies

This parameter is visible only when you set the Stator parameterization parameter to Specify Ls, Lm, and Ms.

Resistance of each of the stator windings.

Option to include or exclude zero-sequence terms.

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

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

Reference point for the rotor 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 rotor and a-phase fluxes are aligned when the rotor 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 torque when the rotor angle is zero.

### Mechanical

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

Rotary damping.

## References

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

[2] Anderson, P. M. Analysis of Faulted Power Systems. Hoboken, NJ: Wiley-IEEE Press, 1995.