Tonći Modrić

3D Computation of the Power Lines Magnetic Field

In this paper, a 3D quasi-static numerical algorithm for computation of the magnetic field produced by power lines is presented. These power lines can be overhead power line phase conductors and shield wires or buried cable line phase conductors. The basis of the presented algorithm is the application of Biot-Savart law and the thin-wire approximation of cylindrical conductors. The catenary form of the power line conductors is approximated by a set of straight cylindrical segments. By summing up contributions of all conductor segments, magnetic field distribution is computed. On the basis of the presented theory, a FORTRAN program PFEMF for computation of the magnetic flux density distribution was developed. For each conductor catenary, it is necessary to define only global coordinates of the beginning and ending points and also the value of the longitudinal phase conductor current. Global coordinates of beginning and ending points of each catenary segment are generated automatically in PFEMF. Numerical results obtained by program PFEMF are compared with results obtained by simple 2D model and results obtained using software package CDEGS.

A surface charge simulation method based on advanced numerical integration

An advanced method for computation of electric field intensity is presented.Subparametric spatial 2D finite elements are developed.Expressions for self and mutual coefficients of 2D finite element nodes are derived.The problem of singularity is solved using advanced double 2D numerical integration. Numerical models for computing low-frequency electromagnetic fields can contain spatial 2D finite elements, which are numerically most demanding due to problem of singularity. In this paper, an advanced time-harmonic quasistatic surface charge simulation method for computation of scalar electric potential and electric field intensity distribution is presented. Subparametric spatial 2D finite elements with an arbitrary number of nodes for description of surface charge density distribution are developed. The problem of singularity that occurs in the double 2D integration over these elements is solved using an originally developed advanced numerical integration based on 2D Gaussian quadrature. Self and mutual coefficients of spatial 2D finite element nodes are numerically computed and included in the system of linear equations for surface charge density distribution computation. The accuracy of the computer program, based on the presented model, is shown in the chosen numerical example with known analytical solution. Numerical model and advanced integration presented herein could be easily extended to non-homogeneous regions and multilayer problems using the image method.

2D computation and measurement of electric and magnetic fields of overhead electric power lines

In this paper a 2D numerical algorithm is presented for the computation of electric and magnetic fields of power lines. The numerical algorithm for the electric field intensity computation takes into account a short power line and approximates the conductor charge density by a constant. The numerical algorithm for magnetic flux density computation is based on the application of the Biot-Savart law. The computed results are compared to measurements taken underneath a 400 kV power line. The computed results and measurements prove to be in good agreements keeping in mind that the sag of the power line section is approximated with a horizontal straight line. Both the computed results as well as measurements confirm the fact that the electric and magnetic fields of high voltage power lines are well within the prescribed limits.

3D Computation of the Overhead Power Lines Electric Field

In this paper, a 3D quasistatic numerical algorithm for computation of the electric field produced by overhead power lines is presented. The real catenary form of the overhead power line phase conductors and shield wires is taken into account with an arbitrary number of straight thin-wire cylindrical segments of active and passive conductors. In order to obtain more precise results of the charge density distribution, segmentation is conducted for each overhead power line span separately. Moreover, the presence of the towers which distort the electric field and significantly reduce its magnitude is taken into account. Therefore, the towers of overhead power lines are approximated using thin-wire cylindrical segments of passive conductors with electric potential equal to zero. From self and mutual coefficients of these components, system of linear equations for computation of the charge density distribution was obtained. In the numerical example, electric field intensity distribution in the vicinity of towers and under the midspan of overhead power lines is shown. In order to verify the accuracy of the presented model, the obtained results are compared with similar published examples and results available in the literature.

Computation of Carson formulas using piecewise quadratic approximation

In this paper, an existing numerical algorithm for high-accurate computation of exact Carson formulas based on piecewise linear approximation is improved. Carson formulas are used for computing of per-unit-length (pul) self and mutual impedances of infinitely long parallel conductors. The proposed algorithm is based on piecewise quadratic approximation of kernel function and analytical integrations of approximated kernel function multiplied by the rest of two integrands. Using proposed algorithm, high-accurate results with desired computer machine n-digit accuracy can be easily obtained. Total number of sample points is significantly decreased with proposed algorithm in comparison with piecewise linear approximation. Results computed by two approximation methods are compared with high-accurate results computed by proposed numerical algorithm for large frequency range.

The difference between voltage and potential difference

In this paper some basic terms such as voltage and potential difference are presented. In many cases they are regarded as identical which leads to confusion with understanding of the fundamental concept of electromagnetic field. Related to this topic, some authors in their books and papers on electromagnetic theory have discussed what the voltmeter actually measures, which is resolved here in a simple way. In this paper it is shown that there is a difference between the terms voltage and potential difference depending on what is the observation point — static fields or time-varying fields. Also in the transmission line model, the voltage between two points depends on the path of integration and, therefore, is ambiguous. What is commonly referred to as voltage, is transversal voltage that is a special case of voltage equal to the potential difference that is unique. Similarly, in electrical circuit analysis, branch voltages are unique and equal to difference of nodal voltages (nodal potentials).