Alpha-Beta-Zero to dq0, dq0 to Alpha-Beta-Zero

(To be removed) Perform transformation from αβ0 stationary reference frame to dq0 rotating reference frame or the inverse

The Specialized Power Systems library will be removed in R2026a. Use the Simscape™ Electrical™ blocks and functions instead. For more information on updating your models, see Upgrade Specialized Power System Models to use Simscape Electrical Blocks.

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Libraries:
Simscape / Electrical / Specialized Power Systems / Control

Description

The Alpha-Beta-Zero to dq0 block performs a transformation of αβ0 Clarke components in a fixed reference frame to dq0 Park components in a rotating reference frame.

The dq0 to Alpha-Beta-Zero block performs a transformation of dq0 Park components in a rotating reference frame to αβ0 Clarke components in a fixed reference frame.

The block supports the two conventions used in the literature for Park transformation:

• Rotating frame aligned with A axis at t = 0. This type of Park transformation is also known as the cosine-based Park transformation.

• Rotating frame aligned 90 degrees behind A axis. This type of Park transformation is also known as the sine-based Park transformation. Use it in Simscape Electrical Specialized Power Systems models of three-phase synchronous and asynchronous machines.

Knowing that the position of the rotating frame is given by ω.t (where ω represents the frame rotation speed), the αβ0 to dq0 transformation performs a −(ω.t) rotation on the space vector U_s = u_α + j· u_β. The homopolar or zero-sequence component remains unchanged.

Depending on the frame alignment at t = 0, the dq0 components are deduced from αβ0 components as follows:

When the rotating frame is aligned with A axis, the following relations are obtained:

$$\begin{array}{l}{U}_s=u_d+j\cdot u_q=\left(u_a+j\cdot u_{\beta }\right)\cdot e^{-j\omega t}\\ \left[\begin{array}{c}{u}_d\\ u_q\\ u_0\end{array}\right]=\left[\begin{array}{ccc}\cos (\omega t)& \sin (\omega t)& 0\\ -\sin (\omega t)& \cos (\omega t)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_a\\ u_{\beta }\\ u_0\end{array}\right]\end{array}$$

The inverse transformation is given by

$$\begin{array}{l}{u}_{\alpha }+j\cdot u_{\beta }=\left(u_d+j\cdot u_q\right)\cdot e^{j\omega t}\\ \left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\\ u_0\end{array}\right]=\left[\begin{array}{ccc}\cos (\omega t)& -\sin (\omega t)& 0\\ \sin (\omega t)& \cos (\omega t)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_d\\ uq\\ u_0\end{array}\right]\end{array}$$

When the rotating frame is aligned 90 degrees behind A axis, the following relations are obtained:

$$\begin{array}{l}{U}_s=u_d+j\cdot u_q=\left(u_{\alpha }+j\cdot u_{\beta }\right)\cdot e^{-j\left(\omega t-\frac{\pi }{2}\right)}\\ \left[\begin{array}{c}{u}_d\\ u_q\\ u_0\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\sin (\omega t)& \sin \left(\omega t-\frac{2\pi }{3}\right)& \sin \left(\omega t+\frac{2\pi }{3}\right)\\ \cos (\omega t)& \cos \left(\omega t-\frac{2\pi }{3}\right)& \cos \left(\omega t+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{u}_a\\ u_b\\ u_c\end{array}\right]\end{array}$$

The inverse transformation is given by

$$u_{\alpha }+j\cdot u_{\beta }=\left(u_d+j\cdot u_q\right)\cdot e^{j\left(\omega t-\frac{\pi }{2}\right)}$$

The abc-to-Alpha-Beta-Zero transformation applied to a set of balanced three-phase sinusoidal quantities u_a, u_b, u_c produces a space vector U_s whose u_α and u_β coordinates in a fixed reference frame vary sinusoidally with time. In contrast, the abc-to-dq0 transformation (Park transformation) applied to a set of balanced three-phase sinusoidal quantities u_a, u_b, u_c produces a space vector U_s whose u_d and u_q coordinates in a dq rotating reference frame stay constant.

Ports

Input

αβ0 — Vectorized αβ0 signal

vector

Vectorized αβ0 signal.

dq0 — Vectorized dq0 signal

vector

Vectorized dq0 signal.

Output

wt — Angular position of dq rotating frame

scalar

The angular position, in radians, of the dq rotating frame relative to the stationary frame.

Parameters

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Rotating frame alignment (at wt=0) — Alignment of rotating frame

90 degrees behind phase A axis (default) | Aligned with phase A axis

Alignment of the rotating frame, when wt = 0, of the dq0 components of a three-phase balanced signal:

$$u_a=\sin (\omega t);u_b=\sin \left(\omega t-\frac{2\pi }{3}\right);u_c=\sin \left(\omega t+\frac{2\pi }{3}\right)$$

(positive-sequence magnitude = 1.0 pu; phase angle = 0 degree)

When you select Aligned with phase A axis, the dq0 components are d = 0, q = −1, and zero = 0.

When you select 90 degrees behind phase A axis, the default option, the dq0 components are d = 1, q = 0, and zero = 0.

Extended Capabilities

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

Version History

Introduced in R2013a