# PMSM

Permanent magnet synchronous motor with sinusoidal flux distribution

**Library:**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.

You can choose different built-in parameterizations for this block. For more information, see the Predefined Parameterization section.

### 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 based on the rotor angle.

For the axes convention, when you set the **Rotor angle
definition** parameter to ```
Angle between the a-phase
magnetic axis and the d-axis
```

, the *a*-phase and
permanent magnet fluxes align when the rotor mechanical angle,
*θ _{r}*, is zero. When you set the

**Rotor angle definition**parameter to

```
Angle
between the a-phase magnetic axis and the q-axis
```

, the rotor
mechanical angle is the angle between the *a*-phase magnetic axis and the rotor

*q*-axis.

### Equations

Voltages across the stator windings are

$$\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]=\left[\begin{array}{ccc}{R}_{s}& 0& 0\\ 0& {R}_{s}& 0\\ 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]+\left[\begin{array}{c}\frac{d{\psi}_{a}}{dt}\\ \frac{d{\psi}_{b}}{dt}\\ \frac{d{\psi}_{c}}{dt}\end{array}\right],$$

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 for the 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

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

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*, and so on, are the mutual inductances of the stator windings._{ba}*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 the rotor electrical angle and defined by

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

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

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

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

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

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

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

where:

*θ*is the rotor mechanical angle._{r}*θ*is the rotor 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.*L*is the stator self-inductance per phase. This value is the average self-inductance of each of the stator windings._{s}*L*is the stator inductance fluctuation. This value is the amount the self-inductance and mutual inductance fluctuate with the changing of the rotor 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

*θ*= 90°. Therefore, the linked motor flux is defined by:

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

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

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

*Nθ*.

_{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:

$$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\\ {v}_{0}\end{array}\right]=P\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]$$

$\left[\begin{array}{c}{i}_{d}\\ {i}_{q}\\ {i}_{0}\end{array}\right]=P\left[\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}\right]$

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

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

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

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

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

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}*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.

### Alternative Flux Linkage Parameterization

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** parameter.

The back EMF constant is the peak voltage induced by the permanent magnet in the per-unit rotational speed of each of the phases. The relationship between the peak permanent magnet flux linkage and the back EMF is:

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

The back EMF, *e _{ph}*, for one phase
is:

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

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

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

When
*L _{d}*=

*L*and the currents in all three phases are balanced, the combined torque

_{q}*T*is:

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

where *I _{pk}* 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
*k _{t}* can also be:

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

where *T* is the measured total torque when
testing with a balanced three-phase current with peak line voltage
*I _{pk}*. The RMS line current is:

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

### Calculating Iron Losses

Iron losses are divided into two terms, one representing the main magnetizing path, and the other representing the cross-tooth tip path that becomes active during field weakened operation. The iron losses model, which is based on the work of Mellor [3].

The term representing the main magnetizing path depends on the induced RMS line-to-neutral stator voltage, $${V}_{{m}_{rms}}^{}$$:

$${P}_{OC}\left({V}_{{m}_{rms}}^{}\right)=\frac{{a}_{h}}{k}{V}_{{m}_{rms}}^{}+\frac{{a}_{j}}{{k}^{2}}{V}_{{m}_{rms}}^{2}+\frac{{a}_{ex}}{{k}^{1.5}}{V}_{{m}_{rms}}^{1.5}$$

This is the dominant term during no-load operation. *k* is the
back emf constant relating RMS volts per Hz. It is defined as $$k={V}_{{m}_{rms}}^{}/f$$, where *f* is the electrical frequency. The first
term on the right-hand side is the magnetic hysteresis loss, the second is the eddy
current loss and the third is the excess loss. The three coefficients appearing on
the numerators are derived from the values that you provide for the open-circuit
hysteresis, eddy, and excess losses.

The term representing the cross-tooth tip path becomes important when a demagnetizing field is set up and can be determined from a finite element analysis short-circuit test. It depends on the RMS emf associated with the cross-tooth tip flux, $${V}_{{d}_{rms}}^{*}$$:

$${P}_{SC}\left({V}_{{d}_{rms}}^{*}\right)=\frac{{b}_{h}}{k}{V}_{{d}_{rms}}^{*}+\frac{{b}_{j}}{{k}^{2}}{V}_{{d}_{rms}}^{*2}+\frac{{b}_{ex}}{{k}^{1.5}}{V}_{{d}_{rms}}^{*1.5}$$

The three numerator terms are derived from the values you provide for the short-circuit hysteresis, eddy, and excess losses.

### Predefined Parameterization

There are multiple available built-in parameterizations for the PMSM block.

This pre-parameterization data allows you to set up the block to represent
components by specific suppliers. The parameterizations of these permanent magnet
synchronous motors match the manufacturer data sheets. To load a predefined
parameterization, double-click the PMSM block, click the **<click to select>** hyperlink of the **Selected
part** parameter and, in the Block Parameterization Manager window,
select the part you want to use from the list of available components.

**Note**

The predefined parameterizations of Simscape™ components use available data sources for the parameter values. Engineering judgement and simplifying assumptions are used to fill in for missing data. As a result, expect deviations between simulated and actual physical behavior. To ensure accuracy, validate the simulated behavior against experimental data and refine component models as necessary.

For more information about pre-parameterization and for a list of the available components, see List of Pre-Parameterized Components.

### Model Thermal Effects

You can expose thermal ports to model the effects of losses that convert power to heat. 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 prior to 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. Nominal
values can come from different sources, one of which is the **Nominal
Values** section in the block dialog box or Property Inspector. For more
information, see System Scaling by Nominal Values.

## Ports

### Conserving

## Parameters

## Model Examples

## 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.

[3] Mellor, P.H., R. Wrobel, and
D. Holliday. “A computationally efficient iron loss model for brushless AC
machines that caters for rated flux and field weakened operation.”
*IEEE Electric Machines and Drives Conference*. May
2009.

## Extended Capabilities

## Version History

**Introduced in R2013b**