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Dq0 TransformThe dq0 transform (often called the Park transform) is a space vector transformation of three-phase time-domain signals from a stationary phase coordinate system (ABC) to a rotating coordinate system (dq0).

The transform applied to time-domain voltages in the natural frame (i.e. ua, ub and uc) is as follows:

Where is the angle between the rotating and fixed coordinate system at each time t and is

an initial phase shift of the voltage.

The inverse transformation from the dq0 frame to the natural abc frame:

As in the Clarke Transform, it is interesting to note that the 0-component above is the same as the zerosequence component in the symmetrical components transform. For example, for voltages Ua, Ub and Uc, the

zero sequence component for both the dq0 and symmetrical components transforms is .

The remainder of this article provides some of the intuition behind why the dq0 transform is so useful in electricalengineering.

Contents [hide]

1 Background2 Classical dq0 Transform in Balanced Systems

2.1 dq0 Transform of Balanced Three-Phase Voltages2.2 dq0 Transform of Balanced Three-Phase Currents2.3 Instantaneous Power in dq0 Frame2.4 Summary of dq0 Transform in Balanced Systems

3 Power Invariant Formulation4 References5 Related Topics

BackgroundThe dq0 transform is essentially an extension of the Clake transform, applying an angle transformation to convertfrom a stationary reference frame to a synchronously rotating frame. The synchronous reference frame can bealigned to rotate with the voltage (e.g. used in voltage source converters) or with the current (e.g. used incurrent source converters).

Historically however, the dq0 transform was introduced earlier than the Clarke transform by R. H. Park in hisseminal 1929 paper on synchronous machine modelling [1].

Classical dq0 Transform in Balanced Systems

dq0 Transform of Balanced Three-Phase VoltagesThe following equations take a two-phase quadrature voltage along the stationary frame and transforms it into atwo-phase synchronous frame (with a reference frame aligned to the voltage):

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Note that in the dq0 frame, the 0-component is the same as that in the frame. Moreover, as we saw in

the Clarke transform, the 0-component is zero for balanced three-phase systems. Therefore in the followingdiscussion on balanced systems, the 0-component will be omitted.

Consider a balanced three-phase voltage with components as follows:

The dq0 transform of this voltage is:

Suppose that we are using a voltage reference frame and will align the synchronous frame with the voltage.Therefore and:

It can be observed that since the synchronous frame is aligned to rotate with the voltage, the d-componentcorresponds to the magnitude of the voltage and the q-component is zero. A plot of the transformation of avoltage from a stationary αβ frame into rotating dq frame is shown in the figure below.

The inverse transform is as follows:

dq0 Transform of Balanced Three-Phase CurrentsThe dq0 transformation can be similarly applied to the current. From a two-phase quadrature stationary (αβ0)current of the form (where δ is the angle at which the current lags the voltage):

We transform it into a two-phase synchronous (dq0) frame:

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This page was last modified on 5 January 2013, at 17:40. About Open Electrical Disclaimers

Instantaneous Power in dq0 FrameThe instantaneous active and reactive power from a set of two-phase (dq) voltages and currents are:

When the synchronous frame is aligned to voltage, we saw earlier that the quadrature component : .

Therefore, the power equations reduce to:

The above equations show that independent control of active and reactive power is possible by means ofcontrolling the dq current components (id and iq).

Summary of dq0 Transform in Balanced SystemsFor three-phase balanced systems, the dq0 transform has the following advantageous characteristics:

1) The dq0 transform reduces three-phase AC quantities (e.g. ua, ub and uc) into two DC quantities (e.g. ud,

uq). For balanced systems, the 0-component is zero. The DC quantities facilitate easier filtering and control.

2) Active and reactive power can be controlled independently by controlling the dq components.

Power Invariant FormulationTBA

Inverse transform:

References[1] R. H. Park, "Two-Reaction Theory of Synchronous Machines: Generalized Method of Analysis - PartI ". Transactions of the AIEE 48: 716–730, 1929

Related TopicsClarke TransformSymmetrical ComponentsReference Frames

Category: Fundamentals

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