d / q transformation

The d / q transformation , also called dq , dq0- and a Park transformation referred serves to three-phase quantities such as in a rotary electric machine with the axes U, V, W in a two-axis coordinate system with the axes d and q to convert . It is part of the mathematical principles for vector control of three-phase machines and describes one of several possible space vector representations . In contrast to the related Clarke transformation , the d / q coordinate system rotates with the rotor in the stationary case and the value pair d / q then represents quantities that are constant over time. The basic form of the d / q transformation was first formulated in 1929 by Robert H. Park .

General

Arrangement of the coils mounted in the stator in the so-called stator-fixed αβ coordinate system.

A three-phase system in the complex plane by three coordinates , and described, which are each offset by an angle of 120 °. They correspond to the three coils of the stationary stator of an induction machine , whereby by definition the axis coincides with the real axis, as shown in the first illustration of the stator-fixed αβ coordinate system of the Clarke transformation. Currents flowing through these coils , and in a symmetrical three-phase system are always 0 in total. ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle U}$ ${\displaystyle I_{U}}$ ${\displaystyle I_{V}}$ ${\displaystyle I_{W}}$

Space vector representation in the d / q coordinate system

With the d / q transformation, the coordinate system with the axes d and q at right angles to each other is rotated with the rotor at the angular frequency , as shown in the second figure. This means that the rotating field at constant speed can be described in the form of two variables d and q that are constant over time. 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 . Changes over time such as speed or torque fluctuations result in changes in d or q over time. The advantage of the transformation is that induction machines can be controlled with a PI controller just as easily as DC machines . ${\displaystyle \Omega _{rotor}}$

In order to allow the d / q coordinate system to rotate with the rotor with the correct angular velocity and phase position, it is necessary to know the exact position in the form of the angle of the rotor. This information, which is essential for the transformation, can be obtained with additional sensors such as Hall or optical sensors, or through feedback such as the evaluation of the electromotive force (EMF) on the stator winding. ${\displaystyle \theta }$

The transformation is not only limited to the electrical currents, but can be applied analogously for all other electrical quantities such as the electrical voltages or the magnetic flux density .

Equations

The amplitude invariant d / q transformation for symmetrical three-phase systems is defined as:

${\displaystyle {\begin{bmatrix}I_{d}\\I_{q}\end{bmatrix}}={\frac {2}{3}}{\begin{bmatrix}\cos(\theta )&\cos(\theta -{\frac {2\pi }{3}})&\cos(\theta -{\frac {4\pi }{3}})\\-\sin(\theta )&-\sin(\theta -{\frac {2\pi }{3}})&-\sin(\theta -{\frac {4\pi }{3}})\end{bmatrix}}{\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}}$

The inverse d / q transformation is:

${\displaystyle {\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}={\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\cos(\theta -{\frac {2\pi }{3}})&-\sin(\theta -{\frac {2\pi }{3}})\\\cos(\theta -{\frac {4\pi }{3}})&-\sin(\theta -{\frac {4\pi }{3}})\end{bmatrix}}{\begin{bmatrix}I_{d}\\I_{q}\end{bmatrix}}}$

If the d / q transformation on the "detour" a previously performed Clarke transformation and the parameters obtained therefrom and carried out, sometimes this path is selected for didactic purposes of teaching, the d / q transformation reduces to a rotation matrix : ${\displaystyle I_{\alpha }}$ ${\displaystyle I_{\beta }}$

${\displaystyle {\begin{bmatrix}I_{d}\\I_{q}\end{bmatrix}}={\begin{bmatrix}\cos(\theta )&\sin(\theta )\\-\sin(\theta )&\cos(\theta )\end{bmatrix}}{\begin{bmatrix}I_{\alpha }\\I_{\beta }\end{bmatrix}}}$

extension

In the case of a three-phase system that is not in equilibrium, the d / q transformation can be extended to the dq0 transformation by adding a third parameter within the framework of the theory of symmetrical components . is the sum of the three phase currents: ${\displaystyle I_{0}}$ ${\displaystyle I_{0}}$

${\displaystyle I_{0}={\frac {1}{3}}(I_{U}+I_{V}+I_{W})}$

with which the three-phase systems that are not in equilibrium can also be described by the amplitude-invariant dq0 transformation:

${\displaystyle {\begin{bmatrix}I_{d}\\I_{q}\\I_{0}\end{bmatrix}}={\frac {2}{3}}{\begin{bmatrix}\cos(\theta )&\cos(\theta -{\frac {2\pi }{3}})&\cos(\theta -{\frac {4\pi }{3}})\\-\sin(\theta )&-\sin(\theta -{\frac {2\pi }{3}})&-\sin(\theta -{\frac {4\pi }{3}})\\{\frac {1}{2}}&{\frac {1}{2}}&{\frac {1}{2}}\end{bmatrix}}{\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}}$

The inverse dq0 transformation is:

${\displaystyle {\begin{bmatrix}I_{U}\\I_{V}\\I_{W}\end{bmatrix}}={\begin{bmatrix}\cos(\theta )&-\sin(\theta )&1\\\cos(\theta -{\frac {2\pi }{3}})&-\sin(\theta -{\frac {2\pi }{3}})&1\\\cos(\theta -{\frac {4\pi }{3}})&-\sin(\theta -{\frac {4\pi }{3}})&1\end{bmatrix}}{\begin{bmatrix}I_{d}\\I_{q}\\I_{0}\end{bmatrix}}}$

The power-invariant dq0 transformation, and thus also the simplified dq transformation, contains the prefactor instead of . The inverse of the inverse transformation must be multiplied by the factor in the performance-invariant case . ${\displaystyle {\sqrt {\frac {2}{3}}}}$ ${\displaystyle {\frac {2}{3}}}$ ${\displaystyle {\sqrt {\frac {2}{3}}}}$

literature

• Padmaraja Yedamale: Field-oriented control without sensors - quieter and more efficient with BLDC motor . In: electronics industry . No. 12 , 2008, p. 38 to 41 .

Individual evidence

1. ↑ RH Park: Two Reaction Theory of Synchronous Machines generalized Method of Analysis Part I . In: AIEE Transactions . Vol. 48, 1929, pp. 716 to 727 , doi : 10.1109 / T-AIEE.1929.5055275 .
2. ↑ A. Binder: Electrical machines and drives . Springer-Verlag, 2012, ISBN 978-3-540-71849-9 , pp. 1016 f .