Implement *abc* to *dq0*
transform

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

The Park Transform block converts the time-domain components of a
three-phase system in an *abc* reference frame to direct, quadrature,
and zero components in a rotating reference frame. The block can preserve the active and
reactive powers with the powers of the system in the *abc* reference
frame by implementing an invariant version of the Park transform. For a balanced system,
the zero component is equal to zero.

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

Alignment of the

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

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

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

$\left[\begin{array}{c}d\\ q\\ 0\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\text{sin}(\theta )& \text{sin}(\theta -\frac{2\pi}{3})& \text{sin}(\theta +\frac{2\pi}{3})\\ \text{cos}(\theta )& \text{cos}(\theta -\frac{2\pi}{3})& \text{cos}(\theta +\frac{2\pi}{3})\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right],$

where:

*a*,*b*, and*c*are the components of the three-phase system in the*abc*reference frame.*d*and*q*are the components of the two-axis system in the rotating reference frame.*0*is the zero component of the two-axis system in the stationary reference frame.

For a power invariant *a*-phase to *q*-axis
alignment, the block implements the transform using this equation:

$\left[\begin{array}{c}d\\ q\\ 0\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\text{sin}(\theta )& \text{sin}(\theta -\frac{2\pi}{3})& \text{sin}(\theta +\frac{2\pi}{3})\\ \text{cos}(\theta )& \text{cos}(\theta -\frac{2\pi}{3})& \text{cos}(\theta +\frac{2\pi}{3})\\ \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$

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

$\left[\begin{array}{c}d\\ q\\ 0\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\text{cos}(\theta )& \text{cos}(\theta -\frac{2\pi}{3})& \text{cos}(\theta +\frac{2\pi}{3})\\ -\text{sin}(\theta )& -\text{sin}(\theta -\frac{2\pi}{3})& -\text{sin}(\theta +\frac{2\pi}{3})\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$

The block implements a power invariant *a*-phase
to *d*-axis alignment as

$\left[\begin{array}{c}d\\ q\\ 0\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\text{cos}(\theta )& \text{cos}(\theta -\frac{2\pi}{3})& \text{cos}(\theta +\frac{2\pi}{3})\\ -\text{sin}(\theta )& -\text{sin}(\theta -\frac{2\pi}{3})& -\text{sin}(\theta +\frac{2\pi}{3})\\ \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}& \sqrt{\frac{1}{2}}\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right].$

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