Implement *dq0* to *abc*
transform

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

The Inverse Park Transform block converts the time-domain direct,
quadrature, and zero components in a rotating reference frame to the components of a
three-phase system in an *a**b**c*
reference frame. The block can preserve the active and reactive powers with the powers
of the system in the rotating 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
*a**b**c* reference frame and a
rotating *d*-*q* 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 *abc* for an:

Alignment of the

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

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

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

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

where:

*d*and*q*are the components of the two-axis system in the rotating reference frame.*a*,*b*, and*c*are the components of the three-phase system in the*a**b**c*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:

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

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

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

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

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

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