Generalized Transformation Matrix for an Unbalanced Electrical System

Dynamic performance of an induction motor suffers due to its complicated control. The interdependence of a phase quantity with other phases makes its control more complex. Hence, a transformation is performed from a stationary to a rotating reference frame to ease the control and improve its dynamics. This process involves a transformation matrix (TM), which includes certain coefficients. The practice mainly focuses on two values for these coefficients to match the amplitude or the power. However, these particular solutions are not the unique values. This work proposes a general solution for the transformation matrix between reference frames. Its impact is mainly profound in the zero-axis component, which is reflected in the case of unbalanced electrical systems. Applicability of the proposed generic transformation is detailed for an induction mo-tor and transmission lines. Further, the validity of the proposed generic transformation matrix is justified through mathematical analysis and simulation.