# Dqo transformation

The d / q transformation, also referred to as a dq, DQ0 and a Park transformation serves to three-phase quantities such as in a three-phase machine with the axes U, V, W in a two-axis coordinate system to transform the d and q axes. It is a part of the mathematical foundations of vector control of three-phase machines and describes one of several possible space vector images. In contrast to related Clarke transformation of the d / q coordinate system rotates in the stationary case with the rotor and the pair of values d / q then provides time-constant variables represents the basic form of the d / q- transformation was first in 1929 by Robert H. Park formulated.

## General

A three-phase system is described in the complex plane by three coordinates, and which are each offset by an angle of 120 °. They correspond to the three coils of the stationary stator of an induction machine, which by definition coincides with the axis of the real axis, as shown in the first illustration of the stator-fixed αβ - coordinate system of the Clarke transformation. Through these coils flowing streams, and are in a symmetrical three- phase system in the sum is always 0

In the d / q transformation, the coordinate system with each other at right angles standing d and q axes rotates along with the circular frequency of the rotor as shown in the second figure. Thus, the rotating field can be described at a constant speed in the form of two time-invariant variables d and q. The value d represents the magnetic flux density of the magnetic excitation in the rotor, and q is an expression for the torque generated by the rotor. Temporal changes as the speed or torque fluctuations result in temporal changes of d and q, respectively. The advantage of the transformation is that induction machines as easy as DC machines can be controlled using a PI controller.

To make the d / q coordinate system co-rotate with the correct phase position and angular velocity of the rotor, it is necessary for the exact location in the form of knowing the angle of the rotor. This is essential for the transformation information may be obtained by additionally attached to the machine sensors such as Hall or optical sensors, or by means of feedback, such as the analysis of the electromotive force ( EMF) of the stator winding.

The transformation is not limited to the electric currents, but may be applied by analogy to all other electrical parameters such as the occurring voltage or the magnetic flux density.

## Equations

The d / q transformation is given as:

The inverse d / q transformation is:

If the d / q- transformation via the "detour " of a previously conducted Clarke transformation and the parameters derived therefrom and carried out, sometimes this path is chosen for didactic purposes in teaching, the d / q- transformation reduces to a rotation matrix:

## Extension

In a three phase system is not in equilibrium can be extended to the DQ0 transformation by adding a third parameter in the theory of symmetrical components of the d / q transformation. is the sum of the three phase currents:

With the then-current for non-equilibrium systems DQ0 three-phase transformation:

And the inverse to dq0 transformation:

The published version of RHPark 1929 has, in contrast to the above power invariant transformation, different pre-factors in the equations.