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Six-phase synchronous machine

**Library:**Simscape / Electrical / Electromechanical / Synchronous

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:

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

$${\theta}_{e}(t)=N{\theta}_{r}(t)+x\_rotor\_offset,$$

where:

*θ*is the rotor electrical angle._{e}*N*is the number of pole pairs.*θ*is the rotor mechanical angle._{r}*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:

*v*,_{d1}*v*, and_{q1}*v*are the_{01}*d*-axis,*q*-axis, and zero-sequence ABC stator voltages, defined by:$$\left[\begin{array}{c}{v}_{d1}\\ {v}_{q1}\\ {v}_{01}\end{array}\right]={P}_{s1}\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right].$$

*v*,_{a}*v*, and_{b}*v*are the ABC stator voltages measured from port_{c}**~ABC**to neutral port**n1**.*ω*is the per-unit base electrical speed._{base}*ψ*,_{d1}*ψ*, and_{q1}*ψ*are the_{01}*d*-axis,*q*-axis, and zero-sequence stator flux linkages for the ABC group.*ω*is the per-unit rotor rotational speed._{r}*R*is the stator resistance._{s}*i*,_{d1}*i*, and_{q1}*i*are the_{01}*d*-axis,*q*-axis, and zero-sequence ABC stator currents, defined by:$$\left[\begin{array}{c}{i}_{d1}\\ {i}_{q1}\\ {i}_{01}\end{array}\right]={P}_{s1}\left[\begin{array}{c}{i}_{A}\\ {i}_{B}\\ {i}_{C}\end{array}\right].$$

*i*,_{a}*i*, and_{b}*i*are the ABC stator currents flowing from port_{c}**~ABC**to port**n1**.

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:

*v*,_{d2}*v*, and_{q2}*v*are the_{02}*d*-axis,*q*-axis, and zero-sequence XYZ stator voltages, defined by:$$\left[\begin{array}{c}{v}_{d2}\\ {v}_{q2}\\ {v}_{02}\end{array}\right]={P}_{s2}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right].$$

*v*,_{x}*v*, and_{y}*v*are the XYZ stator voltages measured from port_{z}**~XYZ**to neutral port**n2**.*ψ*,_{d2}*ψ*, and_{q2}*ψ*are the_{02}*d*-axis,*q*-axis, and zero-sequence stator flux linkages for the XYZ group.*i*,_{d2}*i*, and_{q2}*i*are the_{02}*d*-axis,*q*-axis, and zero-sequence XYZ stator currents, defined by:$$\left[\begin{array}{c}{i}_{d2}\\ {i}_{q2}\\ {i}_{02}\end{array}\right]={P}_{s2}\left[\begin{array}{c}{i}_{{}_{X}}\\ {i}_{Y}\\ {i}_{{}_{Z}}\end{array}\right].$$

*i*,_{x}*i*, and_{y}*i*are the XYZ stator currents flowing from port_{z}**~XYZ**to port**n2**.

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'*is the field winding voltage referred to the stator side._{fd}*v'*and_{kd}*v'*are the_{kq}*dq*-axes damper winding voltages referred to the stator side. They are all equal to 0.*ψ'*,_{fd}*ψ'*, and_{kd}*ψ'*are the magnetic fluxes linking the field circuit, the_{kq}*d*-axis damper winding, and the*q*-axis damper winding.*R'*,_{fd}*R'*, and_{kd}*R'*are the resistances of the rotor field circuit,_{kq}*d*-axis damper winding, and*q*-axis damper winding.*i'*,_{fd}*i'*, and_{kd}*i'*are the field and_{kq}*dq*-axes damper winding currents referred to the stator side.

The stator flux linkage equations are defined by:

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

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

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

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

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

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

where:

*L*is the stator leakage inductance._{l}*L*and_{md}*L*are the mutual inductances of the stator_{mq}*d*-axis and*q*-axis.

The rotor flux linkage equations are defined by:

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

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

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

where:

*L'*is the rotor field winding inductance._{lfd}*L'*is the rotor_{lkd}*d*-axis damper winding inductance.*L'*is the rotor_{lkg}*q*-axis damper winding inductance.

The rotor torque is defined by:

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

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
```

.

[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.

Synchronous Machine Field Circuit | Synchronous Machine Measurement | Synchronous Machine Model 1.0 | Synchronous Machine Model 2.1 | Synchronous Machine Round Rotor | Synchronous Machine Salient Pole