The Method of Symmetrical Components Derived by Reference Frame Theory

In 1965 it was shown that all known real transformations commonly used in the analysis of induction machines were contained in one general change of variables. It is now shown that complex transformations, in particular the method of symmetrical components, are also contained in this same general transformation. Hence all change of variables (transformations), real or complex, commonly used in ac machine analysis stem from one general transformation. Moreover, even without prior knowledge of the existence of symmetrical components, this method naturally evolves from the general transformation as the logical technique to solve unbalanced operation of induction machines. This rigorous approach to the sOblution of unbalanced stator or rotor conditions enables one to appreciate the power and flexibility of the method of symmetrical components in induction machine analysis. The general transformation (arbitrary reference frame) is used to establish the positively and negatively rotating balanced sets for multi-frequency phase variables which need only be periodic. The method of symmetrical components is then derived from the expressions in the arbitrary reference frame. -An open circuited stator phase and unbalanced rotor resistors are used as two specific examples to illustrate the facility of this approach in establishing the symmetrical component voltage equations necessary to analyze these modes of operation.