# Synchronous Machine (Six-Phase)

Six-phase synchronous machine

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## Description

The Synchronous Machine (Six-Phase) block models a six-phase synchronous machine, also known as a dual-star machine.

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

The equivalent circuits of the six-phase synchronous machine for the direct axis, the quadrature axis, and the two zero sequence are:

### Equations

The synchronous machine equations are expressed with respect to a synchronously rotating reference frame, defined by:

`${\theta }_{e}\left(t\right)=N{\theta }_{r}\left(t\right)+x_rotor_offset,$`

where:

• θe is the rotor electrical angle.

• N is the number of pole pairs.

• θr is the rotor mechanical angle.

• x_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.

Two Park transformations map the synchronous machine equations to the rotating reference frame with respect to the electrical angle. The Park transformation for the first group of stator windings, the ABC group, is defined by:

`${P}_{s1}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}{\theta }_{e}& \mathrm{cos}\left({\theta }_{e}-\frac{2\pi }{3}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ -\mathrm{sin}{\theta }_{e}& -\mathrm{sin}\left({\theta }_{e}-\frac{2\pi }{3}\right)& -\mathrm{sin}\left({\theta }_{e}+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$`

The Park transformation for the second group of stator windings, the XYZ group, is defined by:

`${P}_{s2}=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}\left({\theta }_{e}-\frac{\pi }{6}\right)& \mathrm{cos}\left({\theta }_{e}-\frac{5\pi }{6}\right)& \mathrm{cos}\left({\theta }_{e}+\frac{\pi }{2}\right)\\ -\mathrm{sin}\left({\theta }_{e}-\frac{\pi }{6}\right)& -\mathrm{sin}\left({\theta }_{e}-\frac{5\pi }{6}\right)& \mathrm{sin}\left({\theta }_{e}+\frac{\pi }{2}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right].$`

The Park transformations are used to define the per-unit synchronous machine equations.

The stator voltage equations for the ABC group are defined by:

`${v}_{d1}={R}_{s}{i}_{d1}-{\psi }_{q1}{\omega }_{r}+\frac{1}{{\omega }_{base}}\frac{d{\psi }_{d1}}{dt}$`

`${v}_{q1}={R}_{s}{i}_{q1}+{\psi }_{d1}{\omega }_{r}+\frac{1}{{\omega }_{base}}\frac{d{\psi }_{q1}}{dt}$`

`${v}_{01}={R}_{s}{i}_{01}+\frac{1}{{\omega }_{base}}\frac{d{\psi }_{01}}{dt}$`

where:

• vd1, vq1, and v01 are the d-axis, q-axis, and zero-sequence ABC stator voltages, defined by:

`$\left[\begin{array}{c}{v}_{d1}\\ {v}_{q1}\\ {v}_{01}\end{array}\right]={P}_{s1}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$`

va, vb, and vc are the ABC stator voltages measured from port ~ABC to neutral port n1.

• ωbase is the per-unit base electrical speed.

• ψd1, ψq1, and ψ01 are the d-axis, q-axis, and zero-sequence stator flux linkages for the ABC group.

• ωr is the per-unit rotor rotational speed.

• Rs is the stator resistance.

• id1, iq1, and i01 are the d-axis, q-axis, and zero-sequence ABC stator currents, defined by:

`$\left[\begin{array}{c}{i}_{d1}\\ {i}_{q1}\\ {i}_{01}\end{array}\right]={P}_{s1}\left[\begin{array}{c}{i}_{A}\\ {i}_{B}\\ {i}_{C}\end{array}\right].$`

ia, ib, and ic are the ABC stator currents flowing from port to port .

The stator voltage equations for the XYZ group are defined by:

`${v}_{d2}={R}_{s}{i}_{d2}-{\psi }_{q2}{\omega }_{r}+\frac{1}{{\omega }_{base}}\frac{d{\psi }_{d2}}{dt}$`

`${v}_{q2}={R}_{s}{i}_{q2}+{\psi }_{d2}{\omega }_{r}+\frac{1}{{\omega }_{base}}\frac{d{\psi }_{q2}}{dt}$`

`${v}_{02}={R}_{s}{i}_{02}+\frac{1}{{\omega }_{base}}\frac{d{\psi }_{02}}{dt}$`

where:

• vd2, vq2, and v02 are the d-axis, q-axis, and zero-sequence XYZ stator voltages, defined by:

`$\left[\begin{array}{c}{v}_{d2}\\ {v}_{q2}\\ {v}_{02}\end{array}\right]={P}_{s2}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right].$`

vx, vy, and vz are the XYZ stator voltages measured from port ~XYZ to neutral port n2.

• ψd2, ψq2, and ψ02 are the d-axis, q-axis, and zero-sequence stator flux linkages for the XYZ group.

• id2, iq2, and i02 are the d-axis, q-axis, and zero-sequence XYZ stator currents, defined by:

`$\left[\begin{array}{c}{i}_{d2}\\ {i}_{q2}\\ {i}_{02}\end{array}\right]={P}_{s2}\left[\begin{array}{c}{i}_{{}_{X}}\\ {i}_{Y}\\ {i}_{{}_{Z}}\end{array}\right].$`

ix, iy, and iz are the XYZ stator currents flowing from port to port .

The rotor voltage equations are defined by:

`$v{\text{'}}_{fd}=R{\text{'}}_{fd}i{\text{'}}_{fd}+\frac{1}{{\omega }_{base}}\frac{d\psi {\text{'}}_{fd}}{dt}$`

`$v{\text{'}}_{kd}=R{\text{'}}_{kd}i{\text{'}}_{kd}+\frac{1}{{\omega }_{base}}\frac{d\psi {\text{'}}_{kd}}{dt}=0$`

`$v{\text{'}}_{kq}=R{\text{'}}_{kq}i{\text{'}}_{kq}+\frac{1}{{\omega }_{base}}\frac{d\psi {\text{'}}_{kq}}{dt}=0$`

where:

• v'fd is the field winding voltage referred to the stator side.

• v'kd and v'kq are the dq-axes damper winding voltages referred to the stator side. They are all equal to 0.

• ψ'fd, ψ'kd, and ψ'kq are the magnetic fluxes linking the field circuit, the d-axis damper winding, and the q-axis damper winding.

• R'fd, R'kd, and R'kq are the resistances of the rotor field circuit, d-axis damper winding, and q-axis damper winding.

• i'fd, i'kd, and i'kq are the field and dq-axes damper winding currents referred to the stator side.

The stator flux linkage equations are defined by:

`${\psi }_{d1}={L}_{l}{i}_{d1}+{L}_{md}\left({i}_{d1}+{i}_{d2}+{{i}^{\prime }}_{fd}+{{i}^{\prime }}_{kd}\right)$`

`${\psi }_{q1}={L}_{l}{i}_{q1}+{L}_{mq}\left({i}_{q1}+{i}_{q2}+{{i}^{\prime }}_{kq}\right)$`

`${\psi }_{01}={L}_{l}{i}_{01}$`

`${\psi }_{d2}={L}_{l}{i}_{d2}+{L}_{md}\left({i}_{d1}+{i}_{d2}+{{i}^{\prime }}_{fd}+{{i}^{\prime }}_{kd}\right)$`

`${\psi }_{q2}={L}_{l}{i}_{q2}+{L}_{mq}\left({i}_{q1}+{i}_{q2}+{{i}^{\prime }}_{kq}\right)$`

`${\psi }_{02}={L}_{l}{i}_{02}$`

where:

• Ll is the stator leakage inductance.

• Lmd and Lmq are the mutual inductances of the stator d-axis and q-axis.

The rotor flux linkage equations are defined by:

`$\psi {\text{'}}_{fd}=L{\text{'}}_{lfd}i{\text{'}}_{fd}+{L}_{md}\left({i}_{d1}+{i}_{d2}+{{i}^{\prime }}_{fd}+{{i}^{\prime }}_{kd}\right)$`

`$\psi {\text{'}}_{kd}=L{\text{'}}_{lkd}i{\text{'}}_{kd}+{L}_{md}\left({i}_{d1}+{i}_{d2}+{{i}^{\prime }}_{fd}+{{i}^{\prime }}_{kd}\right)$`

`$\psi {\text{'}}_{kq}=L{\text{'}}_{lkq}i{\text{'}}_{kq}+{L}_{mq}\left({i}_{q1}+{i}_{q2}+{{i}^{\prime }}_{kq}\right)$`

where:

• L'lfd is the rotor field winding inductance.

• L'lkd is the rotor d-axis damper winding inductance.

• L'lkg is the rotor q-axis damper winding inductance.

The rotor torque is defined by:

`${T}_{e}={\psi }_{d1}{i}_{q1}-{\psi }_{q1}{i}_{d1}+{\psi }_{d2}{i}_{q2}-{\psi }_{q2}{i}_{d2}.$`

### 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

The Variables settings allow you to specify the priority and initial target values for block variables before simulation. For more information, see Set Priority and Initial Target for Block Variables.

For this block, the Variables settings are visible only if, in the Initial Conditions settings, the Initialization option parameter is set to ```Set targets for rotor angle and Park's transform variables```.

## Ports

### Conserving

expand all

Electrical conserving port associated with the field winding positive terminal.

Electrical conserving port associated with the field winding negative terminal.

Mechanical rotational conserving port associated with the machine rotor.

Mechanical rotational conserving port associated with the machine case.

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`.

Thermal conserving port associated with stator winding a.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

Thermal conserving port associated with stator winding b.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

Thermal conserving port associated with stator winding c.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

Thermal conserving port associated with stator winding x.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

Thermal conserving port associated with stator winding y.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

Thermal conserving port associated with stator winding z.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

Thermal conserving port associated with the rotor.

#### Dependencies

To enable this port, set Modeling option to `Show thermal port`.

## Parameters

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Whether to enable the thermal ports of the block and model the effects of losses that convert power to heat.

### Main

Nominal power.

Nominal root mean square (RMS) line-to-line voltage.

Nominal electrical frequency at which the nominal power is quoted.

Number of machine pole pairs.

Zero-sequence model with two options:

• `Include` — Prioritize model fidelity. An error occurs if you Include zero-sequence terms for simulations that use the Partitioning solver. For more information, see Increase Simulation Speed Using the Partitioning Solver.

• `Exclude` — Prioritize simulation speed for desktop simulation or real-time deployment.

Reference point for the rotor angle measurement. If you select the default value, the rotor and a-phase fluxes are aligned for a zero-rotor angle. Otherwise, an a-phase current generates the maximum torque value for a zero-rotor angle.

### Impedances

Resistance of the stator. This parameter must be greater than `0`.

Leakage inductance of the stator. This parameter must be greater than `0`.

d-axis mutual inductance of the stator. This parameter must be greater than 0.

q-axis mutual inductance of the stator. This parameter must be greater than 0.

Resistance of the field winding of the rotor. This parameter must be greater than 0.

Inductance of the field winding of the rotor. This parameter must be greater than 0.

Resistance of the damper winding of the rotor in the d-axis. This parameter must be greater than 0.

Inductance of the damper winding of the rotor in the d-axis. This parameter must be greater than 0.

Resistance of the damper winding of the rotor in the q-axis. This parameter must be greater than 0.

Inductance of the damper winding of the rotor in the q-axis. This parameter must be greater than 0.

### Initial Conditions

Model for specifying values for certain parameters and variables at the start of simulation:

• To set an operating point independent of the connected network, select ```Set real power, reactive power, terminal voltage and terminal phase```.

• To specify the priority and initial target values for block variables before simulation using the Variables settings, select ```Set targets for rotor angle and Park's transform variables```. For more information, see Set Priority and Initial Target for Block Variables.

Terminal voltage magnitude.

#### Dependencies

To enable this parameter, set Initialization option to ```Set real power, reactive power, terminal voltage, and terminal phase```.

Terminal voltage angle corresponding to the ABC stator.

#### Dependencies

To enable this parameter, set Initialization option to ```Set real power, reactive power, terminal voltage, and terminal phase```.

Terminal active power generated by the stator ABC windings.

#### Dependencies

To enable this parameter, set Initialization option to ```Set real power, reactive power, terminal voltage, and terminal phase```.

Terminal reactive power generated by the stator ABC windings.

#### Dependencies

To enable this parameter, set Initialization option to ```Set real power, reactive power, terminal voltage, and terminal phase```.

Terminal active power generated by the stator XYZ windings.

#### Dependencies

To enable this parameter, set Initialization option to ```Set real power, reactive power, terminal voltage, and terminal phase```.

Terminal reactive power generated by the stator XYZ windings.

#### Dependencies

To enable this parameter, set Initialization option to ```Set real power, reactive power, terminal voltage, and terminal phase```.

### Thermal

To enable these parameters, set Modeling option to `Show thermal port`.

Temperature for which motor parameters are quoted.

Coefficient α in the equation relating resistance to temperature for all three windings, as described in Thermal Model for Actuator Blocks. The default value, `3.93e-3` 1/K, is for copper.

Thermal mass value for each stator winding. The thermal mass is the energy required to raise the temperature by one degree.

Thermal mass of the rotor. The thermal mass is the energy required to raise the temperature of the rotor by one degree.

## References

[1] Kieferndorf, F., Burzanowska , H., Kanerva S., Sario P. "Modeling of rotor based harmonics in dual-star, wound field, synchronous machines." 2008 18th International Conference on Electrical Machines: Vilamoura, 1-6.

[2] Burzanowska , H., Sario P, Stulz C., Joerg P. "Redundant Drive with Direct Torque Control (DTC) and dual-star synchronous machine, simulations and verifications." 2007 European Conference on Power Electronics and Applications: Aalborg, 1-10.

## Version History

Introduced in R2020b