Systematic derivation of clarke and park transformations through vector representation in three-phase systems

english

2025

volume 18, issue 3

Clarke transformation   coordinate transformations   motor control   Park transformation   power electronics education   reference frame theory

Univerza v Mariboru, Fakulteta za energetiko
(Obvezni izvod spletne publikacije)

This paper presents a comprehensive pedagogical treatment of Clarke and Park transformations through systematic vector representation, designed specifically for educational purposes in power electronics and motor control courses. While these fundamental transformations are ubiquitous in modern applications–from field-oriented control of AC machines to grid-connected converter control–educational materials often present them either as matrix operations without geometric foundation, or embedded within machine-specific derivations that obscure the general mathematical structure. We address this pedagogical need by progressing systematically from three-phase voltage equations in the time domain to a spatial vector representation in three-dimensional space, deriving the Clarke transformation through explicit geometric projection onto the plane where balanced quantities reside, and, subsequently, deriving the Park transformation as a time-varying rotation of the Clarke frame. The work establishes an amplitude-invariant formulation, provides an explicit geometric interpretation of the 35.26° angle between coordinate systems, and includes workednumerical examples demonstrating complete transformations. This unified pedagogical treatment bridges the gap between practical application and mathematical foundations, providing educators with a complete geometrically intuitive framework suitable for graduate-level instruction and professional development.

05.03.2026