Clarke's Matrix Correction Procedure for Non Transposed Three-phase Transmission Lines

Abstract

Clarke's matrix has been applied as a phase-mode transformation matrix to three-phase transmission fines substituting the eigenvector matrices. Considering symmetrical untransposed three-phase lines and the frequency range into 10 kHz, some transient simulations have been made with the application of this single real matrix. An actual symmetrical three-phase line on untransposed conditions is associated with Clarke's matrix for error and frequency scan analyses in this paper. Error analyses are calculated for the eigenvalue diagonal elements obtained from Clarke's matrix. The eigenvalue off-diagonal elements from the Clarke's matrix application are compared to the correspondent exact eigenvalues. Based on the characteristic impedance and propagation function values, the frequency scan analyses show that there are great differences between the Clarke's matrix results and the exact ones, considering frequency values from 10 kHz. A correction procedure is applied obtaining two new transformation matrices. These matrices lead to good approximated results when compared to the exact ones. Based on the frequency scan analyses and neglecting the imaginary part of these transformation matrices, similar results to those from new transformation matrices can be obtained. With the correction procedure applied to Clarke's matrix, the relative values of the eigenvalue matrix off-diagonal element obtained from Clarke's matrix are decreased while the frequency scan results are improved.