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# PMSM (Six-Phase)

Six-phase permanent magnet synchronous motor with sinusoidal flux distribution

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• Simscape / Electrical / Electromechanical / Permanent Magnet

• ## Description

The PMSM (Six-Phase) block models a permanent magnet synchronous machine (PMSM) with a six-phase star-wound stator.

A six-phase PMSM has two groups of three-phase stator windings: the ABC group and the XYZ group. These two groups have a 30 degree phase shift.

The figure shows the equivalent electrical circuit for the stator windings. ### Equations

The voltages across the stator windings are defined by:

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

where:

• va, vb, and vc are the individual phase voltages from port ~ABC to neutral port n1.

• vx, vy, and vz are the individual phase voltages from port ~XYZ to neutral port n2.

• Rs is the equivalent resistance of each stator winding.

• ia, ib, and ic are the currents flowing from port ~ABC to port n1.

• ix, iy, and iz are the currents flowing from port ~XYZ to port n2.

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

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

`$\left[\begin{array}{l}{\psi }_{a}\\ {\psi }_{b}\\ {\psi }_{c}\\ {\psi }_{x}\\ {\psi }_{y}\\ {\psi }_{z}\end{array}\right]=\left[\begin{array}{cccccc}{L}_{aa}& {L}_{ab}& {L}_{ac}& {L}_{ax}& {L}_{ay}& {L}_{az}\\ {L}_{ba}& {L}_{bb}& {L}_{bc}& {L}_{bx}& {L}_{by}& {L}_{bz}\\ {L}_{ca}& {L}_{cb}& {L}_{cc}& {L}_{cx}& {L}_{cy}& {L}_{cz}\\ {L}_{xa}& {L}_{xb}& {L}_{xc}& {L}_{xx}& {L}_{xy}& {L}_{xz}\\ {L}_{ya}& {L}_{yb}& {L}_{yc}& {L}_{yx}& {L}_{yy}& {L}_{yz}\\ {L}_{za}& {L}_{zb}& {L}_{zc}& {L}_{zx}& {L}_{zy}& {L}_{zz}\end{array}\right]\left[\begin{array}{l}{i}_{a}\\ {i}_{b}\\ {i}_{c}\\ {i}_{x}\\ {i}_{y}\\ {i}_{z}\end{array}\right]+\left[\begin{array}{l}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\\ {\psi }_{xm}\\ {\psi }_{ym}\\ {\psi }_{zm}\end{array}\right],$`

where:

• ψa, ψb, ψc, ψx, ψy, and ψz are the total fluxes that link each stator winding.

• Laa, Lbb, Lcc, Lxx, Lyy, and Lzz 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, ψxm, ψym, and ψzm are the permanent magnet fluxes linking the stator windings.

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

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

`$\begin{array}{l}{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\frac{\pi }{3}\right)\right)\\ {L}_{cc}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+2\frac{\pi }{3}\right)\right)\\ {L}_{xx}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-\frac{\pi }{6}\right)\right)\\ {L}_{yy}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-5\frac{\pi }{6}\right)\right)\\ {L}_{zz}={L}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+\frac{\pi }{2}\right)\right)\\ {L}_{ab}={L}_{ba}=-{M}_{s}-{L}_{m}\left(\text{2cos}\left({\theta }_{e}+\frac{\pi }{6}\right)\right)\\ {L}_{bc}={L}_{cb}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(\text{2}\left({\theta }_{e}+\frac{\pi }{6}-2\frac{\pi }{3}\right)\right)\\ {L}_{ca}={L}_{ac}=-{M}_{s}-{L}_{m}\text{cos}\left({\theta }_{e}+\frac{\pi }{6}+2\frac{\pi }{3}\right)\\ {L}_{xy}={L}_{yx}=-{M}_{s}-{L}_{m}\mathrm{cos}\left(2{\theta }_{e}\right)\\ {L}_{yz}={L}_{zy}=-{M}_{s}-{L}_{m}\text{cos}\left(2\left({\theta }_{e}-2\frac{\pi }{3}\right)\right)\\ {L}_{zx}={L}_{xz}=-{M}_{s}-{L}_{m}\text{cos}\left(2\left({\theta }_{e}+2\frac{\pi }{3}\right)\right)\\ {L}_{ax}={L}_{xa}=\sqrt{3}{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-\frac{\pi }{12}\right)\right)\\ {L}_{ay}={L}_{ya}=-\sqrt{3}{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-5\frac{\pi }{12}\right)\right)\\ {L}_{az}={L}_{za}={L}_{m}\text{cos}\left(2\left({\theta }_{e}+\frac{\pi }{4}\right)\right)\\ {L}_{bx}={L}_{xb}={L}_{m}\text{cos}\left(2\left({\theta }_{e}-5\frac{\pi }{12}\right)\right)\\ {L}_{by}={L}_{yb}=\sqrt{3}{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+\frac{\pi }{4}\right)\right)\\ {L}_{bz}={L}_{zb}=-\sqrt{3}{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-\frac{\pi }{12}\right)\right)\\ {L}_{cx}={L}_{xc}=-\sqrt{3}{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}+\frac{\pi }{4}\right)\right)\\ {L}_{cy}={L}_{yc}={L}_{m}\text{cos}\left(2\left({\theta }_{e}-\frac{\pi }{12}\right)\right)\\ {L}_{cz}={L}_{zc}=\sqrt{3}{M}_{s}+{L}_{m}\text{cos}\left(2\left({\theta }_{e}-5\frac{\pi }{12}\right)\right)\end{array}$`

where:

• θr is the rotor mechanical angle.

• θe is the rotor electrical angle.

• 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.

• 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:

`$\left[\begin{array}{l}{\psi }_{am}\\ {\psi }_{bm}\\ {\psi }_{cm}\\ {\psi }_{xm}\\ {\psi }_{ym}\\ {\psi }_{zm}\end{array}\right]=\left[\begin{array}{l}{\psi }_{m}\mathrm{cos}{\theta }_{e}\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{e}-2\pi /3\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{e}+2\pi /3\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{e}-\pi /6\right)\\ {\psi }_{m}\mathrm{cos}\left({\theta }_{e}-5\pi /6\right)\\ {\psi }_{m}cos\left({\theta }_{e}+\pi /2\right)\end{array}\right],$`

where ψm is the permanent magnet flux linkage.

### Simplified Electrical Equations

Applying a decoupled transformation to the block electrical equations produces an expression for torque that is independent of the rotor angle.

The decoupled transformation is defined by:

`$P\left({\theta }_{e}\right)=\frac{1}{3}\left[\begin{array}{cccccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-2\frac{\pi }{3}\right)& \mathrm{cos}\left({\theta }_{e}+2\frac{\pi }{3}\right)& cos\left({\theta }_{e}-\frac{\pi }{6}\right)& \mathrm{cos}\left({\theta }_{e}-5\frac{\pi }{6}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{\pi }{2}\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-2\frac{\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}+2\frac{\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}-\frac{\pi }{6}\right)& -\mathrm{sin}\left({\theta }_{e}-5\frac{\pi }{6}\right)& -\mathrm{sin}\left({\theta }_{e}+\frac{\pi }{2}\right)\\ 1& -\frac{1}{2}& -\frac{1}{2}& -\frac{\sqrt{3}}{2}& \frac{\sqrt{3}}{2}& 0\\ 0& -\frac{\sqrt{3}}{2}& \frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{1}{2}& -1\\ 1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\end{array}\right].$`

The transformation matrix, P, has this pseudo-orthogonal property:

`${P}^{-1}\left({\theta }_{e}\right)=3{P}^{T}\left({\theta }_{e}\right).$`

Using the decoupled transformation on the stator winding voltages and currents transforms them to the dq0 frame, which is independent of the rotor angle.

To obtain the d-axis, q-axis, and zero-sequence stator voltages and flux linkages for the ABC and XYZ groups, apply the transformation to the voltage and flux linkage equations:

`$\left[\begin{array}{l}{v}_{d}\\ {v}_{q}\\ {v}_{z1}\\ {v}_{z2}\\ {v}_{01}\\ {v}_{02}\end{array}\right]=\left[\begin{array}{cccccc}{R}_{s}& 0& 0& 0& 0& 0\\ 0& {R}_{s}& 0& 0& 0& 0\\ 0& 0& {R}_{s}& 0& 0& 0\\ 0& 0& 0& {R}_{s}& 0& 0\\ 0& 0& 0& 0& {R}_{s}& 0\\ 0& 0& 0& 0& 0& {R}_{s}\end{array}\right]\left[\begin{array}{l}{i}_{d}\\ {i}_{q}\\ {i}_{z1}\\ {i}_{z2}\\ {i}_{01}\\ {i}_{02}\end{array}\right]+\left[\begin{array}{c}-{\psi }_{q}\\ {\psi }_{d}\\ 0\\ 0\\ 0\\ 0\end{array}\right]N\omega +\frac{d}{dt}\left[\begin{array}{c}{\psi }_{d}\\ {\psi }_{q}\\ {\psi }_{z1}\\ {\psi }_{z2}\\ {\psi }_{01}\\ {\psi }_{02}\end{array}\right]$`

`$\left[\begin{array}{c}{\psi }_{d}\\ {\psi }_{q}\\ {\psi }_{z1}\\ {\psi }_{z2}\\ {\psi }_{01}\\ {\psi }_{02}\end{array}\right]=\left[\begin{array}{cccccc}{L}_{d}& 0& 0& 0& 0& 0\\ 0& {L}_{q}& 0& 0& 0& 0\\ 0& 0& {L}_{0}& 0& 0& 0\\ 0& 0& 0& {L}_{0}& 0& 0\\ 0& 0& 0& 0& {L}_{0}& 0\\ 0& 0& 0& 0& 0& {L}_{0}\end{array}\right]\left[\begin{array}{l}{i}_{d}\\ {i}_{q}\\ {i}_{z1}\\ {i}_{z2}\\ {i}_{01}\\ {i}_{02}\end{array}\right]+\left[\begin{array}{c}{\psi }_{m}\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]$`

where:

• vd, vq, vz1, vz2, v01, and v02 are the d, q, z1, and z2 components and zero-sequence stator voltages for the ABC and XYZ groups, defined by:

`$\left[\begin{array}{l}{v}_{d}\\ {v}_{q}\\ {v}_{z1}\\ {v}_{z2}\\ {v}_{01}\\ {v}_{02}\end{array}\right]=P\left[\begin{array}{l}{v}_{a}\\ {v}_{b}\\ {v}_{c}\\ {v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right].$`

• id, iq, iz1, iz2, i01, and i02 are the d-axis, q-axis, and zero-sequence stator currents for the ABC and XYZ groups, defined by:

`$\left[\begin{array}{l}{i}_{d}\\ {i}_{q}\\ {i}_{z1}\\ {i}_{z2}\\ {i}_{01}\\ {i}_{02}\end{array}\right]=P\left[\begin{array}{l}{i}_{a}\\ {i}_{b}\\ {i}_{c}\\ {i}_{x}\\ {i}_{y}\\ {i}_{z}\end{array}\right].$`

• ${L}_{d}={L}_{s}+4{M}_{s}+3{L}_{m}$ is the stator d-axis inductance.

• ${L}_{q}={L}_{s}+4{M}_{s}-3{L}_{m}$ is the stator q-axis inductance.

• ${L}_{0}={L}_{s}-2{M}_{s}$ is the stator zero-sequence inductance.

• ω is the rotor mechanical rotational speed.

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

The torque equation is defined by:

`$T=3N\left[{i}_{q}\left({i}_{d}{L}_{d}+{\psi }_{m}\right)-{i}_{d}{i}_{q}{L}_{q}\right].$`

### 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

Three-phase electrical port associated with the stator ABC windings.

Three-phase electrical port associated with the stator XYZ windings.

Electrical conserving port associated with the neutral point of the ABC winding configuration.

#### Dependencies

To enable this port, set Zero sequence to `Include`.

Electrical conserving port associated with the neutral point of the XYZ winding configuration.

#### Dependencies

To enable this port, set 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

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 parameterization to ```Specify flux linkage```.

Torque constant with any of the stator windings.

#### Dependencies

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

Back EMF constant with any of the stator windings.

#### Dependencies

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

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 results in an error for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.

• `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 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 the mechanical translational port R. The value can be zero.

Rotary damping.

 Krause, Paul, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek, eds. Analysis of electric machinery and drive systems. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2013. https://doi.org/10.1002/9781118524336.

 Su, Jian Yong, Jin Bo Yang, and Gui Jie Yang. Research on Vector Control and PWM Technique of Six-Phase PMSM. Advanced Materials Research 516–517 (May 2012): 1626–31. https://doi.org/10.4028/www.scientific.net/AMR.516-517.1626.

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