Journal
IEEE TRANSACTIONS ON ENERGY CONVERSION
Volume 34, Issue 4, Pages 2070-2083Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TEC.2019.2941175
Keywords
Transmission line matrix methods; Harmonic analysis; Symmetric matrices; Rotors; History; Couplings; Synchronous machines; dq0; Park; Clarke; reference-frame theory; locus diagram; power quality
Funding
- Khalifa University under the flagship project [8474000026, TEC-00200-2019]
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Transformations between abc, stationary dq0 (alpha beta 0) and rotating dq0 reference-frames are used extensively in the analysis and control of three-phase technologies such as machines and inverters. Previous work on deriving the matrices describing these transformations follows one of two approaches. The first approach derives Clarke's matrix by modifying symmetrical components. Park's matrix can be subsequently found from a rotation matrix. The second approach derives Park's matrix using trigonometric projection by interpreting the transformation as a rotation in the plane of the cross-section of a machine. Then, Clarke's matrix can be found trivially using a reference angle of zero in Park's matrix. This paper presents a third approach to deriving the Clarke and Park transformation matrices: a geometric interpretation. The approach exploits properties of the linear transformation using the Cartesian representation. We introduce the locus diagram of a three-phase quantity and show how these diagrams have applications in power quality. We show that, unlike a phasor diagram, a single locus diagram can fully represent a three-phase system with harmonics.
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