# PMSM (Four-Phase)

Four-phase permanent magnet synchronous motor with sinusoidal flux distribution

*Since R2022a*

**Libraries:**

Simscape /
Electrical /
Electromechanical /
Permanent Magnet

## Description

The PMSM (Four-Phase) block models a permanent magnet synchronous motor with a four-phase cross-wound or square-wound stator. The figure shows the equivalent electrical circuits for the cross-connected and square-connected stator windings.

### Motor Construction

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

The voltages across the stator windings are

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

where:

*v*,_{a}*v*,_{b}*v*, and_{c}*v*are the individual phase voltages across the stator windings._{d}*R*is the equivalent resistance of each stator winding._{s}*i*,_{a}*i*,_{b}*i*, and_{c}*i*are the currents flowing in the stator windings._{d}$$\frac{d{\psi}_{a}}{dt},$$$$\frac{d{\psi}_{b}}{dt},$$$$\frac{d{\psi}_{c}}{dt},$$ and $$\frac{d{\psi}_{d}}{dt}$$ are the rates of change for the magnetic flux in each stator winding.

The permanent magnet and the four windings contribute to the total flux linking each winding. The total flux is

$$\left[\begin{array}{l}{\psi}_{a}\\ {\psi}_{b}\\ {\psi}_{c}\\ {\psi}_{d}\end{array}\right]=\left[\begin{array}{cccc}{L}_{a}{}_{a}& {L}_{a}{}_{b}& {L}_{a}{}_{c}& {L}_{a}{}_{d}\\ {L}_{ba}& {L}_{bb}& {L}_{b}{}_{c}& {L}_{b}{}_{d}\\ {L}_{c}{}_{a}& {L}_{c}{}_{b}& {L}_{c}{}_{c}& {L}_{c}{}_{d}\\ {L}_{d}{}_{a}& {L}_{d}{}_{b}& {L}_{d}{}_{c}& {L}_{d}{}_{d}\end{array}\right]\left[\begin{array}{l}{i}_{a}\\ {i}_{b}\\ {i}_{c}\\ {i}_{d}\end{array}\right]+\left[\begin{array}{l}{\psi}_{am}\\ {\psi}_{bm}\\ {\psi}_{cm}\\ {\psi}_{dm}\end{array}\right],$$

where:

*ψ*,_{a}*ψ*,_{b}*ψ*, and_{c}*ψ*are the total fluxes that link each stator winding._{d}*L*,_{aa}*L*,_{bb}*L*, and_{cc}*L*are the self-inductances of the stator windings._{dd}*L*,_{ab}*L*,_{ac}*L*, and so on, are the mutual inductances of the stator windings._{ba}*ψ*,_{am}*ψ*,_{bm}*ψ*, and_{cm}*ψ*are the permanent magnet fluxes linking the stator windings._{dm}

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

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

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

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

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

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

${L}_{ab}={L}_{ba}=-{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta}_{e}+\pi /4\right)\right)$

$${L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\text{cos}\left(\text{2}\left({\theta}_{e}+\pi /4-\pi /2\right)\right)$$

${L}_{cd}={L}_{dc}=-{M}_{s}-{L}_{m}\text{cos}\left(2\left({\theta}_{e}+\pi /4+\pi \right)\right)$

${L}_{ad}={L}_{da}=-{M}_{s}-{L}_{m}\text{cos}\left(2\left({\theta}_{e}+\pi /4+\pi /2\right)\right)$

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

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

where:

*θ*is the rotor mechanical angle._{r}*θ*is the rotor electrical angle._{e}*rotor offset*is`-pi/2`

if you define the rotor electrical angle with respect to the*d*-axis, or`0`

if you define the rotor electrical angle with respect to the*q*-axis.*L*is the stator per-phase 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 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-a'* is at maximum
when *θ _{e}* = 0° and zero when

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

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

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

### Simplified Electrical Equations

To remove the rotor angle dependence for the inductive terms, you perform a
transformation, *T*, on the motor equations.

The *T* transformation is defined by:

$$T({\theta}_{e})=\frac{1}{2}\left[\begin{array}{cccc}\mathrm{cos}{\theta}_{e}& \mathrm{cos}({\theta}_{e}-\pi /2)& \mathrm{cos}({\theta}_{e}+\pi )& \mathrm{cos}({\theta}_{e}+\pi /2)\\ -\mathrm{sin}{\theta}_{e}& -\mathrm{sin}({\theta}_{e}-\pi /2)& -\mathrm{sin}({\theta}_{e}+\pi )& -\mathrm{sin}({\theta}_{e}+\pi /2)\\ 1/\sqrt{2}& -1/\sqrt{2}& 1/\sqrt{2}& -1/\sqrt{2}\\ 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}\end{array}\right],$$

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

*Nθ*.

_{r}*N*is the number of pole pairs.

The transformation matrix has this pseudo-orthogonal property:

$${T}^{-1}({\theta}_{e})=2{T}^{t}({\theta}_{e}).$$

Using the *T* transformation on the stator winding voltages and
currents transforms them to the dq0 and xy frames, which are independent of the
rotor angle:

$$\left[\begin{array}{l}{v}_{ds}\\ {v}_{qs}\\ {v}_{x}\\ {v}_{0}\end{array}\right]=T\left[\begin{array}{l}{v}_{a}\\ {v}_{b}\\ {v}_{c}\\ {v}_{d}\end{array}\right]$$

$$\left[\begin{array}{l}{i}_{ds}\\ {i}_{qs}\\ {i}_{x}\\ {i}_{0}\end{array}\right]=T\left[\begin{array}{l}{i}_{a}\\ {i}_{b}\\ {i}_{c}\\ {i}_{d}\end{array}\right]$$

Applying this transformation to the first two electrical equations produces these equations

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

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

${v}_{x}={R}_{s}{i}_{x}+\frac{{L}_{d}+{L}_{q}}{2}\frac{d{i}_{x}}{dt}$

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

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

where:

*L*=_{d}*L*+_{s}*M*+ 2_{s}*L*._{m}*L*is the stator_{d}*d*-axis inductance.*L*=_{q}*L*+_{s}*M*− 2_{s}*L*._{m}*L*is the stator_{q}*q*-axis inductance.*L*=_{0}*L*– 3_{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.

### Alternative Flux Linkage Parameterization

You can parameterize the motor by 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 four phases are balanced, the combined torque

_{q}*T*is:

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

where *I _{pk}* is the peak current in any of
the four windings.

The factor 2 is calculated from the steady-state sum of the torques from all
phases. Therefore, the torque constant *k _{t}*
can also be:

$${k}_{t}=\frac{1}{2}\left(\frac{T}{{I}_{pk}}\right),$$

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

$${k}_{t}=\sqrt{\frac{1}{2}}\left(\frac{T}{{i}_{line,rms}}\right).$$

### 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 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

## Ports

### Conserving

## Parameters

## References

[1] Della Torre, F., et al.
*Symmetrical Components and Space-Vector Transformations for Four-Phase
Networks*. IEEE Transactions on Power Delivery, vol. 23, no. 4,
pp.2191-200, Oct., 2008.

[2] Anderson, P. M.
*Analysis of Faulted Power Systems*. IEEE Press Power Systems
Engineering Series, 1995. ISBN 0-7803-1145-0.

## Extended Capabilities

## Version History

**Introduced in R2022a**