Dqo transformationIn electrical engineering, direct–quadrature–zero (or dq0 or dqo) transformation or zero–direct–quadrature (or 0dq or odq) transformation is a mathematical transformation used to simplify the analysis of three-phase circuits. In the case of balanced three-phase circuits, application of the dqo transform reduces the three AC quantities to two DC quantities. Simplified calculations can then be carried out on these imaginary DC quantities before performing the inverse transform to recover the actual three-phase AC results. It is often used in order to simplify the analysis of three-phase synchronous machines or to simplify calculations for the control of three-phase inverters. The dqo transform presented here is exceedingly similar to the transform first proposed in 1929 by Robert H. Park.[1] In fact, the dqo transform is often referred to as Park’s transformation.

Definition

The dqo transform applied to three-phase currents is shown below in matrix form:[2]

The inverse transform is:

Geometric Interpretation

The dqo transformation can be thought of in geometric terms as the projection of the three separate sinusoidal phase quantities onto two axes rotating with the same angular velocity as the sinusoidal phase quantities. The two axes are called the direct, or d, axis; and the quadrature or q, axis; that is, with the q-axis being at an angle of 90 degrees from the direct axis.

Shown above is the dqo transform as applied to the stator of a synchronous machine. There are three windings separated by 120 physical degrees. The three phase currents are equal in magnitude and are separated from one another by 120 electrical degrees. The three phase currents lag their corresponding phase voltages by . The d-q axis is shown rotating with angular velocity equal to , the

same angular velocity as the phase voltages and currents. The d axis makes an angle with the A winding which has been chosen

as the reference.The currents and are constant DC quantities.

Comparison with other transforms

Park's transformation

The transformation originally proposed by Park differs slightly from the one given above. Park's transformation is:

and

Although useful, Park's transformation is not power invariant whereas the dqo transformation defined above is[2]:88. Park's transformation gives the same zero component as the method of symmetrical components. The dqo transform shown above gives a zero

component which is larger than that of Park or symmetrical components by a factor of .

transformαβγ

The dqo transform is conceptually similar to the transformαβγ . Whereas the dqo transform is the projection of the phase quantities onto a rotating two-axis reference frame, the transform can be thought of as the projection of the phase quantities onto a stationaryαβγ two-axis reference frame

References

In-line references

1. ̂ R.H. Park Two Reaction Theory of Synchronous Machines AIEE Transactions 48:716-730 (1929).2. ^ a b P.M. Anderson and A.A. Fouad Power System Control and Stability IEEE Press (2003). ISBN 978-81-265-1818-0

General references

J. Lewis Blackburn Symmetrical Components for Power Systems Engineering, Marcel Dekker, New York (1993). ISBN 0-8247-8767-6 Zhang et al. A three-phase inverter with a neutral leg with space vector modulation IEEE APEC '97 Conference Proceedings (1997).

Symmetrical componentsIn electrical engineering, the method of symmetrical components is used to simplify analysis of unbalanced three phase power systems under both normal and abnormal conditions.

Description

In 1918 Charles Legeyt Fortescue presented a paper[1] which demonstrated that any set of N unbalanced phasors (that is, any such polyphase signal) could be expressed as the sum of N symmetrical sets of balanced phasors, for values of N that are prime. Only a single frequency component is represented by the phasors.

In a three-phase system, one set of phasors has the same phase sequence as the system under study (positive sequence; say ABC), the second set has the reverse phase sequence (negative sequence; ACB), and in the third set the phasors A, B and C are in phase with each other (zero sequence). Essentially, this method converts three unbalanced phases into three independent sources, which makes asymmetric fault analysis more tractable.

By expanding a one-line diagram to show the positive sequence, negative sequence and zero sequence impedances of generators, transformers and other devices including overhead lines and cables, analysis of such unbalanced conditions as a single line to ground short-circuit fault is greatly simplified. The technique can also be extended to higher order phase systems.

Physically, in a three phase winding a positive sequence set of currents produces a normal rotating field, a negative sequence set produces a field with the opposite rotation, and the zero sequence set produces a field that oscillates but does not rotate between phase windings. Since these effects can be detected physically with sequence filters, the mathematical tool became the basis for the design of protective relays, which used negative-sequence voltages and currents as a reliable indicator of fault conditions. Such relays may be used to trip circuit breakers or take other steps to protect electrical systems.

The analytical technique was adopted and advanced by engineers at General Electric and Westinghouse and after World War II it was an accepted method for asymmetric fault analysis.

The three-phase case

Symmetrical components are most commonly used for analysis of three-phase electrical power systems. If the phase quantities are expressed in phasor notation using complex numbers, a vector can be formed for the three phase quantities. For example, a vector for three phase voltages could be written as

where the subscripts 0, 1, and 2 refer respectively to the zero, positive, and negative sequence components. The sequence components differ only by their phase angles, which are symmetrical and so are radians or 120°. Define the operator phasor vector forward by that angle.

Note that α3 = 1 so that α−1 = α2.

The zero sequence components are in phase; denote them as:

and the other phase sequences as:

Thus,

where

Conversely, the sequence components are generated from the analysis equations

where

An intuitive feeling

The phasors form a closed triangle (e.g., outer voltages or line to line voltages). To find the synchronous and inverse components of the phases, take any side of the outer triangle and draw the two possible equilateral triangles sharing the selected side as base. These two equilateral triangles represent a synchronous and inverse system. If the phasors V were a perfectly synchronous system, the vertex of the outer triangle not on the base line would be at the same position as the corresponding vertex of the equilateral triangle representing the synchronous system. Any amount of inverse component would mean a deviation from this position. The deviation is exactly 3 times the inverse phase component. The synchronous component is in the same manner 3 times the deviation from the "inverse equilateral triangle". The directions of these components are correct for the relevant phase. It seems counter intuitive that this works for all three phases regardless of the side chosen but that is the beauty of this illustration.

For an illustration see Napoleon's Theorem.

Poly-phase Case

It can be seen that the transformation matrix above is a discrete Fourier transform, and as such, symmetrical components can be calculated for any poly-phase system. However, by Pontryagin duality, only certain groups have a unique inverse, which is necessary for use in fault analysis.

See also

Symmetry

References

1. ̂ Charles L. Fortescue, "Method of Symmetrical Co-Ordinates Applied to the Solution of Polyphase Networks". Presented at the 34th annual convention of the AIEE (American Institute of Electrical Engineers) in Atlantic City, N.J. on 28 July 1918. Published in: AIEE Transactions, vol. 37, part II, pages 1027-1140 (1918). For a brief history of the early years of symmetrical component theory, see: J. Lewis Blackburn, Symmetrical Components for Power Engineering (Boca Raton, Florida: CRC Press, 1993), pages 3-4.

J. Lewis Blackburn Symmetrical Components for Power Systems Engineering, Marcel Dekker, New York (1993). ISBN 0-8247-8767-6 William D. Stevenson, Jr. Elements of Power System Analysis Third Edition, McGraw-Hill, New York (1975). ISBN 0-07-061285-4. History article from IEEE on early development of symmetrical components, retrieved May 12, 2005. Westinghouse Corporation, Applied Protective Relaying, 1976, Westinghouse Corporation, no ISBN, Library of Congress card no. 76-8060 - a

standard reference on electromechanical protective relays

Alpha–beta transformation

In electrical engineering, the alpha-beta ( ) transformation (also known as the Clarke transformation) is a mathematical transformation employed to simplify the analysis of three-phase circuits. Conceptually it is similar to the dqo transformation. One very

useful application of the transformation is the generation of the reference signal used for space vector modulation control of three-phase inverters.

Contents

[hide]

1 Definition 2 Geometric Interpretation

o 2.1 transform 3 References 4 See also

Definition

The transform applied to three-phase currents, as used by Edith Clarke, is shown below in matrix form:[1]

The inverse transform is:

Alternatively, the scaling of the transform can be chosen to instead of , then the inverse transformation matrix is also scaled

by .[2]

In a balanced system and thus and two of the phase currents suffice to compute the and components. In this case the transform simplifies to[3]

and

Geometric Interpretation

The transformation can be thought of as the projection of the three phase quantities (voltages or currents) onto two stationary axes, the alpha axis and the beta axis.

Shown above is the transform as applied to three symmetrical currents flowing through three windings separated by 120 physical degrees. The

three phase currents lag their corresponding phase voltages by . The - axis is shown with the axis aligned with phase 'A'. The current vector

rotates with angular velocity . There is no component since the currents are balanced.

transform

The transform is conceptually similar to the transform. Whereas the dqo transform is the projection of the phase quantities onto a rotating two-axis reference frame, the transform can be thought of as the projection of the phase quantities onto a stationary two-axis reference frame.

References

1. ̂ W. C. Duesterhoeft, Max W. Schulz and Edith Clarke (july 1951). "Determination of Instantaneous Currents and Voltages by Means of Alpha, Beta, and Zero Components". Transactions of the American Institute of Electrical Engineers 70 (2): 1248–1255. doi:10.1109/T-AIEE.1951.5060554. ISSN 0096-3860.

2. ̂ S. CHATTOPADHYAY, M. MITRA, S. SENGUPTA (2008). "Area Based Approach for Three Phase Power Quality Assessment in Clarke Plane". Journal of Electrical Systems (01): 62. Retrieved 2012-04-26.

3. ̂ F. Tahri, A.Tahri, Eid A. AlRadadi and A. Draou Senior, "Analysis and Control of Advanced Static VAR compensator Based on the Theory of the Instantaneous Reactive Power," presented at ACEMP, Bodrum, Turkey, 2007.