# PMSM (Five-Phase)

Five-phase permanent magnet synchronous motor with sinusoidal flux distribution

• Library:
• Simscape / Electrical / Electromechanical / Permanent Magnet

## Description

The PMSM (Five-Phase) block models a permanent magnet synchronous motor with a five-phase star-wound stator. The figure shows the equivalent electrical circuit for the star-connected stator windings.

You can also model the permanent magnet synchronous motor either in a pentagon-wound or a pentacle-wound configuration by setting Winding type to `Pentagon-wound` or `Pentacle-wound`.

### 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 the rotor mechanical angle, θr, is zero. The block supports a second rotor axis definition in which the rotor mechanical angle is defined as the angle between the a-phase magnetic axis and the rotor q-axis.

### Equations

The voltages across the stator windings are defined by:

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

where:

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

• Rs is the equivalent resistance of each stator winding.

• ia, ib, ic, id, and ie are the currents flowing in the stator windings.

• $\frac{d{\psi }_{a}}{dt},$$\frac{d{\psi }_{b}}{dt},$$\frac{d{\psi }_{c}}{dt}$ $\frac{d{\psi }_{d}}{dt},$ and $\frac{d{\psi }_{e}}{dt}$ are the rates of change of magnetic flux in each stator winding.

The permanent magnet and the five windings contribute to the total flux linking each winding. The total flux is defined by:

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

where:

• ψa, ψb, ψc, ψd, and ψe are the total fluxes that link each stator winding.

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

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

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

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

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

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

`${L}_{bb}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-2\pi /5\right)\right),$`

`${L}_{cc}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-4\pi /5\right)\right),$`

`${L}_{dd}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+4\pi /5\right)\right),$`

`${L}_{ee}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+2\pi /5\right)\right),$`

`${L}_{ab}={L}_{ba}={L}_{ce}={L}_{ec}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}-2\pi /5\right),$`

`${L}_{bc}={L}_{cb}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}-6\pi /5\right),$`

`${L}_{cd}={L}_{dc}={L}_{be}={L}_{eb}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}\right),$`

`${L}_{de}={L}_{ed}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}+6\pi /5\right),$`

`${L}_{ea}={L}_{ae}={L}_{bd}={L}_{db}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}+2\pi /5\right),$`

`${L}_{ac}={L}_{ca}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}-4\pi /5\right),$`

and

`${L}_{ad}={L}_{da}=-{M}_{s}+{L}_{m}\text{cos}\left(2{\theta }_{e}+4\pi /5\right),$`

where:

• θr is the rotor mechanical angle.

• θe is the rotor electrical angle.

• 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.

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

• Lm is the stator inductance fluctuation. This value is 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-a' is at maximum when θe = 0° and zero when θe = 90°. Therefore, the linked motor flux is defined by:

`$\left[\begin{array}{l}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\\ {\psi }_{dm}\\ {\psi }_{em}\end{array}\right]=\left[\begin{array}{l}{\psi }_{m}\mathrm{cos}{\theta }_{e}\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{r}-2\pi /5\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{r}-4\pi /5\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{r}+4\pi /5\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{r}+2\pi /5\right)\end{array}\right],$`

where ψm 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:

`$T\left({\theta }_{e}\right)=\frac{2}{5}\left[\begin{array}{ccccc}\mathrm{sin}{\theta }_{e}& \mathrm{sin}\left({\theta }_{e}-2\pi /5\right)& \mathrm{sin}\left({\theta }_{e}-4\pi /5\right)& \mathrm{sin}\left({\theta }_{e}+4\pi /5\right)& \mathrm{sin}\left({\theta }_{e}+2\pi /5\right)\\ \mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-2\pi /5\right)& \mathrm{cos}\left({\theta }_{e}-4\pi /5\right)& \mathrm{cos}\left({\theta }_{e}+4\pi /5\right)& \mathrm{cos}\left({\theta }_{e}+2\pi /5\right)\\ \mathrm{sin}{\theta }_{e}& \mathrm{sin}\left({\theta }_{e}+4\pi /5\right)& \mathrm{sin}\left({\theta }_{e}-2\pi /5\right)& \mathrm{sin}\left({\theta }_{e}+2\pi /5\right)& \mathrm{sin}\left({\theta }_{e}-4\pi /5\right)\\ \mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}+4\pi /5\right)& \mathrm{cos}\left({\theta }_{e}-2\pi /5\right)& \mathrm{cos}\left({\theta }_{e}+2\pi /5\right)& \mathrm{cos}\left({\theta }_{e}-4\pi /5\right)\\ 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}& 1/\sqrt{2}\end{array}\right],$`

where θe is the electrical angle defined as r. N is the number of pole pairs.

The transformation matrix has the following pseudo-orthogonal property:

`${T}^{-1}\left({\theta }_{e}\right)=\frac{5}{2}{T}^{t}\left({\theta }_{e}\right).$`

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:

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

and

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

Applying this transformation to the first two electrical equations produces the following equations that define the block behavior:

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

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

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

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

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

and

`$T=\frac{5}{2}N\left({i}_{qs}\left({i}_{ds}{L}_{d}+{\psi }_{m}\right)-{i}_{ds}{i}_{qs}{L}_{q}\right),$`

where:

• Ld = Ls + Ms + 5/2 Lm. Ld is the stator d-axis inductance.

• Lq = Ls + Ms − 5/2 Lm. Lq is the stator q-axis inductance.

• L0 = Ls – 4Ms. L0 is the stator zero-sequence inductance.

• 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 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 five phases are balanced, it follows that the combined torque T is given by:

`$T=\frac{5}{2}{k}_{t}{i}_{q}=\frac{5}{2}{k}_{t}{I}_{pk},$`

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

The factor 5/2 is calculated from the steady-state sum of the torques from all phases. Therefore the torque constant kt could also be defined as:

`${k}_{t}=\frac{2}{5}\left(\frac{T}{{I}_{pk}}\right),$`

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

`${k}_{t}=\sqrt{\frac{2}{5}}\left(\frac{T}{{i}_{line,rms}}\right).$`

### 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.

## Ports

### Conserving

expand all

Electrical conserving port associated with a-phase.

Electrical conserving port associated with b-phase.

Electrical conserving port associated with c-phase.

Electrical conserving port associated with d-phase.

Electrical conserving port associated with e-phase.

Electrical conserving port associated with the neutral phase.

#### Dependencies

To enable this port, set Winding Type to `Star-wound` and Zero sequence to `Include`.

Mechanical rotational conserving port associated with the motor rotor.

Mechanical rotational conserving port associated with the motor case.

## Parameters

expand all

### Main

Configuration for the windings:

• `Star-wound` — The windings are star-wound.

• `Pentagon-wound` — The windings are pentagon-wound. The a-phase is connected between ports a and b, the b-phase between ports b and c, the c-phase between ports c and d, the d-phase between ports d and e, and the e-phase between ports e and a.

• `Pentacle-wound` — The windings are pentacle-wound. The a-phase is connected between ports a and d, the b-phase between ports b and e, the c-phase between ports c and a, the d-phase between ports d and b, and the e-phase between ports e and c.

Number of permanent magnet pole pairs on the rotor.

Permanent magnet flux linkage, specified as ```Specify flux linkage```, ```Specify torque constant```, or ```Specify back EMF constant```.

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

#### Dependencies

To enable this parameter, set Permanent magnet flux linkage to `Specify flux linkage`.

Torque constant with any of the stator windings.

#### Dependencies

To enable this parameter, set Permanent magnet flux linkage to `Specify torque constant`.

Back EMF constant with any of the stator windings.

#### Dependencies

To enable this parameter, set Permanent magnet flux linkage to `Specify back EMF constant`.

Stator parameterization, specified as `Specify Ld, Lq, and L0` or `Specify Ls, Lm, and Ms`.

d-axis inductance.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq, and L0```.

q-axis inductance.

#### Dependencies

To enable this parameter, set Stator parameterization to ```Specify Ld, Lq, and L0```.

Zero-sequence inductance.

#### Dependencies

To enable this parameter either:

• Set Winding Type to `Star-wound`, Zero sequence to `Include`, and Stator parameterization to `Specify Ld, Lq, and L0`.

• Set Winding Type to `Pentagon-wound`.

• Set Winding Type to `Pentacle-wound`.

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

#### Dependencies

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

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

#### Dependencies

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

Average mutual inductance between the stator windings.

#### Dependencies

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

Resistance of each of the stator windings.

Whether 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.

#### Dependencies

To enable this parameter, set the Winding Type to `Star-wound`.

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] L. Parsa and H. A. Toliyat. Sensorless Direct Torque Control of Five-Phase Interior Permanent-Magnet Motor Drives. IEEE Transactions on Industry Applications, vol. 43, no. 4, pp.952-959, July–Aug., 2007.

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