Voltage and Current Mode Vector Analyses of Correction Procedure Application to Clarke’s Matrix—Symmetrical Three-Phase Cases

Clarke’s matrix has been applied as a phase-mode transformation matrix to three-phase transmission lines substituting the eigenvector matrices. Considering symmetrical untransposed three-phase lines, 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 to 1 MHz. A correction procedure is applied obtaining two new transformation matrices. These matrices lead to good approximated results when compared to the exact ones. 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. The steps of correction procedure application are detailed, investigating the in-fluence of each step on the obtained two new phase-mode transformation matrices