Step by step analyses of Clarke´s matrix correction procedure for untransposed three-phase transmission line cases

The correction procedure for Clarke´s matrix, considering three-phase transmission line analyzes, is analyzed step by step in this paper, searching to improve the application of this procedure. Changing the eigenvectors as modal transformation matrices, Clarke´s matrix has been applied to analyses for transposed and untransposed three-phase transmission line cases. It is based on the fact that Clarke´s matrix is an eigenvector matrix for transposed three-phase transmission lines considering symmetrical and asymmetrical cases. Because of this, the application of this matrix has been analyzed considering untransposed three-phase transmission lines. In most of these cases, the errors related to the eigenvalues can be considered negligible. It is not true when it is analyzed the elements that are not in main diagonal of the quasi-mode matrix. This matrix is obtained from the application of Clarke´s matrix. The quasi-mode matrix is correspondent to the eigenvalue matrix. Their off-diagonal elements represent couplings among the quasi-modes. So, the off-diagonal quasi-mode element relative values are not negligible when compared to the eigenvalues that correspond to the coupled quasi-modes. Minimizing these relative values, the correction procedure is analyzed in detail, checking some alternatives for the correction procedure application.