Modal Transformation Obtained from Clarke´s Matrix - Asymmetrical Three-phase case

Some constant matrices can be used as phase-mode transformation matrices for transposed three-phase transmission lines. Clarke´s matrix is one of these options. Its application as a phase-mode transformation matrix for untransposed three-phase transmission lines has been analyzed through error and frequency scan comparisons. Based on an actual untransposed asymmetrical three-phase transmission line example, a correction procedure is applied searching for better results from the Clarke´s matrix application as a phase-mode transformation matrix. The error analyses are carried out using Clarke´s matrix and the new transformation matrices obtained from the correction procedure. Applying Clarke´s matrix, the relative errors of the eigenvalue matrix elements can be considered negligible and the relative values of the off-diagonal elements are significant. If the the corrected transformation matrices are used, the relative values of the off-diagonal elements are decreased. Based on the results of these analyses, the homopolar mode is more sensitive to the frequency influence than the two other modes related to three-phase lines.