Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

Implement *dq0* to *αβ0*
transform

**Library:**Simscape / Electrical / Control / Mathematical Transforms

The Park to Clarke Angle Transform block converts the direct, quadrature, and zero components in a rotating reference frame to alpha, beta, and zero components in a stationary reference frame. For balanced systems, the zero components are equal to zero.

You can configure the block to align the phase *a*-axis of the
three-phase system to either the *q*- or *d*-axis of
the rotating reference frame at time, *t* = 0. The figures show the
direction of the magnetic axes of the stator windings in the three-phase system, a
stationary *αβ0* reference frame, and a rotating *dq0*
reference frame where:

The

*a*-axis and the*q*-axis are initially aligned.The

*a*-axis and the*d*-axis are initially aligned.

In both cases, the angle *θ* =
*ω**t*, where

*θ*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.*ω*is the rotational speed of the*d*-*q*reference frame.*t*is the time, in s, from the initial alignment.

The figures show the time-response of the individual components of equivalent balanced
*dq0* and *αβ0* for an:

Alignment of the

*a*-phase vector to the*q*-axisAlignment of the

*a*-phase vector to the*d*-axis

The Park to Clarke Angle Transform block implements the transform
for an *a*-phase to *q*-axis alignment as

$\left[\begin{array}{c}\alpha \\ \beta \\ 0\end{array}\right]=\left[\begin{array}{ccc}\text{sin}(\theta )& \text{cos}(\theta )& 0\\ -\text{cos}(\theta )& \text{sin}(\theta )& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}d\\ q\\ 0\end{array}\right]$

where:

*d*and*q*are the direct-axis and quadrature-axis components of the two-axis system in the rotating reference frame.*0*is the zero component.*α*and*β*are the alpha-axis and beta-axis components of the two-phase system in the stationary reference frame.

For an *a*-phase to *d*-axis alignment, the
block implements the transform using this equation:

$\left[\begin{array}{c}\alpha \\ \beta \\ 0\end{array}\right]=\left[\begin{array}{ccc}\text{cos}(\theta )& -\text{sin}(\theta )& 0\\ \text{sin}(\theta )& \text{cos}(\theta )& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}d\\ q\\ 0\end{array}\right]$

[1] Krause, P., O. Wasynczuk, S. D. Sudhoff, and S. Pekarek. *Analysis of
Electric Machinery and Drive Systems.* Piscatawy, NJ: Wiley-IEEE Press,
2013.