Eigenvalue analyses of two parallel lines using a single real transformation matrix

For a typical non-symmetrical system with two parallel three phase transmission lines, modal transformation is applied using some examples of single real transformation matrices. These examples are applied searching an adequate single real transformation matrix to two parallel three phase transmission line systems. The analyses are started with the eigenvector and eigenvalue studies, using Clarke's transformation or linear combinations of Clarke's elements. The Z C and parameters are analyzed for the case that presents the smallest errors between the exact eigenvalues and the single real transformation matrix application results. The single real transformation determined for this case is based on Clarke's matrix and its main characteristic is the use of a unique homopolar reference. So, the homopolar mode becomes a connector mode between the two three-phase circuits of the analyzed system. ©2005 IEEE.

Non-transposed three-phase line analyses with a single real transformation matrix

In transmission line transient analyses, a single real transformation matrix can obtain exact modes when the analyzed line is transposed. For non-transposed lines, the results are not exact. In this paper, non-symmetrical and non transposed three-phase line samples are analyzed with a single real transformation matrix application (Clarke's matrix). Some interesting characteristics of this matrix application are: single, real, frequency independent, line parameter independent, identical for voltage and current determination. With Clarke's matrix use, mathematical simplifications are obtained and the developed model can be applied directly in programs based on time domain. This model works without convolution procedures to deal with phase-mode transformation. In EMTP programs, Clarke's matrix can be represented by ideal transformers and the frequency dependent line parameters can be represented by modified-circuits. With these representations, the electrical values at any line point can be accessed for phase domain or mode domain using the Clarke matrix or its inverse matrix. For symmetrical and non-transposed lines, the model originates quite small errors. In addition, the application of the proposed model to the non-symmetrical and non-transposed three phase transmission lines is investigated. ©2005 IEEE.

Modal representation of three-phase lines applying two transformation matrices: Evaluation of its eigenvectors

The objective of this paper is to show an alternative representation in time domain of a non-transposed three-phase transmission line decomposed in its exact modes by using two transformation matrices. The first matrix is Clarke's matrix that is real, frequency independent, easily represented in computational transient programs (EMTP) and separates the line into Quasi-modes α, β and zero. After that, Quasi-modes a and zero are decomposed into their exact modes by using a modal transformation matrix whose elements can be synthesized in time domain through standard curve-fitting techniques. The main advantage of this alternative representation is to reduce the processing time because a frequency dependent modal transformation matrix of a three-phase line has nine elements to be represented in time domain while a modal transformation matrix of a two-phase line has only four elements. This paper shows modal decomposition process and eigenvectors of a non-transposed three-phase line with a vertical symmetry plane whose nominal voltage is 440 kV and line length is 500 km. ©2006 IEEE.

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.

An alternative procedure to decrease the dimension of the frequency dependent modal transformation matrices: Application in three-phase transmission lines with a vertical symmetry plane

The objective of this paper is to show an alternative representation in time domain of a non-transposed three-phase transmission line decomposed in its exact modes by using two transformation matrices. The first matrix is Clarke's matrix that is real, frequency independent, easily represented in computational transient programs (EMTP) and separates the line into Quasi-modes α, β and zero. After that, Quasi-modes α and zero are decomposed into their exact modes by using a modal transformation matrix whose elements can be synthesized in time domain through standard curve-fitting techniques. The main advantage of this alternative representation is to reduce the processing time because a frequency dependent modal transformation matrix of a three-phase line has nine elements to be represented in time domain while a modal transformation matrix of a two-phase line has only four elements. This paper shows modal decomposition process and eigenvectors of a non-transposed three-phase line with a vertical symmetry plane whose nominal voltage is 440 kV and line length is 500 km. © 2006 IEEE.

Production and heat treatment of cast graphitic steels with additions of niobium

The optimal combination of the mechanical characteristics of austempered spheroidal graphitic cast steel together with modern casting techniques yielded an economically promising product. The maximum potential of the usage of these steels is related to fabrication and characterization techniques, among which, one of the most important is the cooling diagram (TTT curve). In this work, 3 heats of graphitic steels with the following nominal compositions were cast: 1.0 % C, 2.3 % Si, 0.4 % Mn, and with niobium contents of. 0.0 %, 0.5 % and 1.0 %. TTT curves were determined by dilatometric testing and test specimens of these steels were austempered. The samples were then characterized by hardness testing and optical and SEM microscopy. Tensile, impact (no notch) and wear testing were also performed. The addition of niobium produced significant alterations in the TTT diagrams. Increasing niobium content moves the pearlite transformation nose to the right and the bainitic transformation nose to the left. Tensile strength of these alloys was high, in the range of 1700 MPa and impact values were around of 45 Joules for alloy with 1 % Nb, 49 Joules for alloy with 0.5 % Nb and fracture did not occur for the alloy without the addition of Nb.

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.

Production and heat treatment of cast graphitic steels with additions of niobium

The optimal combination of the mechanical characteristics of austempered spheroidal graphitic cast steel together with modern casting techniques yielded an economically promising product. The maximum potentiality of the usage of these steels is related to fabrication and characterization techniques, among which, one of the most important is the cooling diagram (TTT curve). In this work, 3 heats of graphitic steels with the following nominal compositions were cast: 1.0%C, 2.3%Si, 0.4% Mn, and with niobium contents of 0.0%, 0.5% and 1.0%. TTT curves were determined by dilatometric testing and test specimens of these steels were austempered. The samples were then characterized by hardness testing and optical and SEM microscopy. Tensile, impact (no notch) and wear testing were also performed. The addition of niobium produced significant alterations in the TTT diagrams. Increasing niobium content moves the pearlite transformation nose to the right and the bainitic transformation nose to the left. Tensile strength of these alloys was high, in the range of 1700 MPa and impact values were around of 45 Joules for alloy with 1% Nb, 49 Joules for alloy with 0.5% Nb and fracture did not occur for the alloy without the addition of Nb.

Fiscal decentralization: increasing social cohesion among widely disparate territorial units

Includes bibliography

Education and knowledge: basic pillars of changing production patterns with social equity

Includes bibliography

Structural reforms, productivity and technological change in Latin America

Includes bibliography