Single real transformation matrices applied to double three-phase transmission lines

Single real transformation matrices are tested as phase-mode transformation matrices of typical symmetrical systems with double three-phase and two parallel double three-phase transmission lines. These single real transformation matrices are achieved from eigenvector matrices of the mentioned systems and they are based on Clarke's matrix. Using linear combinations of the Clarke's matrix elements, the techniques applied to the single three-phase lines are extended to systems with 6 or 12 phase conductors. For transposed double three-phase lines, phase Z and Y matrices are changed into diagonal matrices in mode domain. Considering non-transposed cases of double three-phase lines, the results are not exact and the error analyses are performed using the exact eigenvalues. In case of two parallel double three-phase lines, the exact single real transformation matrix has not been obtained yet. Searching for this exact matrix, the analyses are based on a single homopolar reference. For all analyses in this paper, the homopolar mode is used as the only homopolar reference for all phase conductors of the studied system. (C) 2008 Elsevier B.V. All rights reserved.

Collisional semiclassical approximations in phase-space representation

The Gaussian wave-packet phase-space representation is used to show that the expansion in powers of a of the quantum Liouville propagator leads, in the zeroth-order term, to results close to those obtained in the statistical quasiclassical method of Lee and Scully in the Weyl-Wigner picture. It is also verified that, propagating the Wigner distribution along the classical trajectories, the amount of error is less than that coming from propagating the Gaussian distribution along classical trajectories.

A THEORETICAL RESULT ON PHASE-LAG

We prove a Theoretical result which relates phase-lag to the error constant of a numerical method for periodic problem y'' = f(x, y).

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.

Time Domain Analyses for Three-phase Lines with Corrected Modal Transformation Matrix

The results presented in this paper are based on a research about the application of approximated transformation matrices for electromagnetic transient analyses and simulations in transmission lines. Initially, it has developed the application of a single real transformation matrix for a double three-phase transmission lines, because the symmetry of the distribution of the phase conductors and the ground wires. After this, the same type of transformation matrix has applied for symmetrical single three-phase transmission lines. Analyzing asymmetrical single three-phase lines, it has used three different line configurations. For these transmission line types, the errors between the eigenvalues and the approximated results, called quasi modes, have been considered negligible. on the other hand, the quasi mode eigenvalue matrix for each case was not a diagonal one. and the relative values of the off-diagonal elements of the approximated quasi mode matrix are not negligible, mainly for the low frequencies. Based on this problem, a correction procedure has been applied for minimizing the mentioned relative values. For the correction procedure application, symmetrical and asymmetrical single three-phase transmission line samples have been used. Checking the correction procedure results, analyses and simulations have been carried out in mode and time domain. In this paper, the last results of mentioned research are presented and they related to the time domain simulations.

Correction procedure applied to a single real transformation matrix -Untransposed three-phase transmission line cases

Clarke's matrix has been used as an eigenvector matrix for transposed three-phase transmission lines and it can be applied as a phase-mode transformation matrix for transposed cases. Considering untransposed three-phase transmission lines, Clarke's matrix is not an exact eigenvector matrix. In this case, the errors related to the diagonal elements of the Z and Y matrices can be considered negligible, if these diagonal elements are compared to the exact elements in domain mode. The mentioned comparisons are performed based on the error and frequency scan analyses. From these analyses and considering untransposed asymmetrical three-phase transmission lines, a correction procedure is determined searching for better results from the Clarke's matrix use as a phase-mode transformation matrix. Using the 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. Applying the corrected transformation matrices, the relative values of the off-diagonal elements are decreased. The comparisons among the results of these analyses show that the homopolar mode is more sensitive to the frequency influence than the two other modes related to three-phase lines. © 2006 IEEE.

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 applicaton 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. © 2007 IEEE.