Decoupled Transform (Six-Phase, Dual-Star)

R2026b

Implement abc and xyz to dqz₁z₂o₁o₂ transform with variable angular displacement

Renamed from Decoupled Transform (Six-Phase) in R2026b

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Decoupled Transform (Six-Phase, Dual-Star) block

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Libraries:
Simscape / Electrical / Control / Mathematical Transforms

Description

The Decoupled Transform (Six-Phase, Dual-Star) block converts the time-domain components of a six-phase system in abc and xyz reference frames to direct, quadrature, z₁, z₂, o₁, and o₂ components in a rotating reference frame. The three-phase systems abc and xyz are spatially shifted by β electrical degrees.

You can configure the block to align the phase a-axis of the six-phase system to either the d-axis or q-axis of the rotating reference frame at time t equal to 0.

θ = ωt + Offset is the angle between the a and q axes for the q-axis alignment or the angle between the a and d axes for the d-axis alignment, where:

• ω is the rotational speed of the rotational reference frame.

• t is the time, in s, from the initial alignment.

• $$Offset=\begin{array}{c}0,d-axisalignment\\ -\frac{\pi }{2},q-axisalignment\end{array}$$

Equations

The Decoupled Transform (Six-Phase, Dual-Star) block implements the transform as

$$\left[\begin{array}{c}d\\ q\\ z_1\\ z_2\\ o_1\\ o_2\end{array}\right]=\frac{1}{3}\left[\begin{array}{cccccc}\cos \theta & \cos \left(\theta -\frac{2\pi }{3}\right)& \cos \left(\theta +\frac{2\pi }{3}\right)& \cos \left(\theta -\beta \right)& \cos \left(\theta -\frac{2\pi }{3}-\beta \right)& \cos \left(\theta +\frac{2\pi }{3}-\beta \right)\\ -\sin \theta & -\sin \left(\theta -\frac{2\pi }{3}\right)& -\sin \left(\theta +\frac{2\pi }{3}\right)& -\sin \left(\theta -\beta \right)& -\sin \left(\theta -\frac{2\pi }{3}-\beta \right)& -\sin \left(\theta +\frac{2\pi }{3}-\beta \right)\\ \cos \theta & \cos \left(\theta -\frac{2\pi }{3}\right)& \cos \left(\theta +\frac{2\pi }{3}\right)& -\cos \left(\theta -\beta \right)& -\cos \left(\theta -\frac{2\pi }{3}-\beta \right)& -\cos \left(\theta +\frac{2\pi }{3}-\beta \right)\\ -\sin \theta & -\sin \left(\theta -\frac{2\pi }{3}\right)& -\sin \left(\theta +\frac{2\pi }{3}\right)& \sin \left(\theta -\beta \right)& \sin \left(\theta -\frac{2\pi }{3}-\beta \right)& \sin \left(\theta +\frac{2\pi }{3}-\beta \right)\\ 1& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 1& 1\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\\ x\\ y\\ z\end{array}\right].$$

Ports

Input

abc — a-, b-, and c-phase components

vector

Components of the three-phase system in an abc reference frame, specified as a vector.

Data Types: single | double

xyz — x-, y-, and z-phase components

vector

Components of the three-phase system in an xyz reference frame, specified as a vector.

Data Types: single | double

θ_elec — Rotational angle

scalar

Angular position of the rotating reference frame, specified as a scalar.

Data Types: single | double

Output

dqz1z2o1o2 — Direct, quadrature, z₁, z₂, o₁, and o₂ components

vector

Direct, quadrature, z₁-, z₂-, o₁-, and o₂-axis components of the system in the rotating reference frame, returned as a vector.

Data Types: single | double

Parameters

Angular displacement (degrees) — Angular displacement between abc and xyz three-phase systems

30 (default) | scalar

Since R2026b

Angular displacement between the abc and xyz three-phase systems, in degrees.

Phase-a axis alignment — dqz₁z₂o₁o₂ reference frame alignment

Q-axis (default) | D-axis

Whether to align the a-phase vector of the abc reference frame to the d-axis or q-axis of the rotating reference frame.

References

[1] Amy, John. "Six Phase Synchronous Machine Model in Machine Variables and “Dq” Variables". 2019 IEEE Electric Ship Technologies Symposium (ESTS), Washington, DC, USA, 2019, pp. 421-429. DOI.org (Crossref), https://doi.org/10.1109/ESTS.2019.8847819.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2021a

R2026b: Specify angular displacement between abc and xyz three-phase systems

You can specify the angular displacement between abc and xyz three-phase systems by using the Angular displacement (degrees) parameter.

Before R2026b, the abc and xyz three-phase systems were always shifted by 30 degrees.

R2026b: Renamed from Decoupled Transform (Six-Phase)

The Decoupled Transform (Six-Phase, Dual-Star) block has been renamed from Decoupled Transform (Six-Phase).

This change has no compatibility impact. When you open an existing model that contains a Decoupled Transform (Six-Phase) block, the software automatically updates your model to use the Decoupled Transform (Six-Phase, Dual-Star) block.