Systematic Derivation of Clarke and Park Transformations through Vector Representation in Three-Phase Systems
DOI:
https://doi.org/10.18690/jet.18.3.145-160.2025Keywords:
Clarke transformation, Park transformation, coordinate transformations, power electronics education, motor control, reference frame theoryAbstract
This paper presents a comprehensive mathematical derivation of Clarke and Park transformations from first principles through systematic vector representation. Despite over a century of widespread application in power electronics and motor control, rigorous derivations of these fundamental transformations remain scattered across literature. We address this gap by systematically progressing from three-phase voltage equations in time domain to spatial vector representation in three-dimensional space, then deriving Clarke transformation through geometric projection onto the plane where balanced quantities reside. Subsequently, we derive Park transformation as time-varying rotation of the Clarke frame. The work establishes both amplitude-invariant and power-invariant formulations, explains the geometric significance of the 35.26° angle between coordinate systems, and reveals the mathematical basis for zero-sequence component extraction. This unified treatment bridges the gap between these transformations' ubiquitous practical application and their fundamental mathematical origins.
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