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Three-phase and two-phase stationary reference frames

Clarke TransformThe Clarke or transform is a space vector transformation of time-domain signals (e.g. voltage, current,

flux, etc) from a natural three-phase coordinate system (ABC) into a stationary two-phase reference frame (). It is named after electrical engineer Edith Clarke [1].

Consider the voltage phasors in the figure to theright. In the natural reference frame, the voltagedistribution of the three stationary axes Ua, Ub,

and Uc are 120o apart from each other.Cartesian axes are also portrayed, where Uα is

the horizontal axis aligned with phase Ua, and

the vertical axis rotated by 90o is indicated byUβ. Uα and Uβ have the same magnitude in per

unit.

Three-phase voltages varying in time along theaxes a, b, and c, can be algebraicallytransformed into two-phase voltages, varying intime along the axes α and β by the followingtransformation matrix:

The inverse transformation can also be obtainedto transform the quantities back from two-phase to three-phase:

It is interesting to note that the 0-component in the Clarke transform is the same as the zero sequencecomponent in the symmetrical components transform. For example, for voltages Ua, Ub and Uc, the zero

sequence component for both the Clarke and symmetrical components transforms is .

Contents [hide]

1 Clarke Transform of Balanced Three-Phase Voltages2 Clarke Transform of Balanced Three-Phase Currents3 References4 Related Topics

Clarke Transform of Balanced Three-Phase VoltagesConsider the following balanced three-phase voltage waveforms:

Taking the Clarke transform, we get:

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Time domain simulation result of transformation from three-phase stationary into two-phase stationarycoordinated system is shown in the following figures:

Three-phase voltages in the time domain

Transformation of three-phase voltages into two-phase orthogonalvoltages

From the equations and figures above, it can be concluded that in the balanced condition, Uα is a sine function,

Uβ is a cosine function and U0 is zero.

Clarke Transform of Balanced Three-Phase CurrentsSimilarly, one can calculate the Clarke transform of balanced three-phase currents (which lags the voltage by anarbitrary angle δ):

Using the same procedure as before, the Clarke transform is:

We can see that as in the voltage case, Iα is a sine function, Iβ is a cosine function and U = I0 is zero. However

note the lagging phase angle δ.

References[1] Edith Clarke, "Circuit Analysis of AC Power Systems. Vol. I.", Wiley, New York, 1943

Related Topics

Clarke Transform - Open Electrical 10/21/2013

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This page was last modified on 22 July 2013, at 05:41. About Open Electrical Disclaimers

dq0 TransformSymmetrical ComponentsReference Frames

Category: Fundamentals

Clarke Transform - Open Electrical 10/21/2013

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