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Consider the propagation of electromagnetic energy through multiconductor three-phase high-voltage transmission line with arbitrary number of conductors. The mathematical formulation of the problem represents the system of partial differential equations known as transmission line equations. When constructing the capacitance matrix of an electric line, it is necessary to solve the boundary problems for the Laplace equation with nonlocal boundary conditions. The aim of this research is to develop a new method for constructing a potential coefficients matrix, which allows us to express the voltage vector in the wires through the charge vector. The application of this method leads to the necessity of solving a boundary value problem for Laplace equations with nonlocal boundary conditions. A new type of non-local boundary conditions has been identified, which has not been previously investigated. The boundary conditions of this type are represented by the contour integral over the boundary of the region from an unknown potential. In the general case, the obtained problem does not have a unique solution. The theorem of existence and uniqueness of the solution for the boundary value problem for the Laplace equation with nonlocal boundary conditions is proved. However, the requirement that the potential on the wire surface is constant allows us to prove the existence and uniqueness of the solution for considered problem. In order to illustrate the application of the developed approach we have solved numerically the problem for the sector-shaped cable with three cores. The values of potential and capacitive coefficients obtained by calculation are given. The error of the values of the diagonal elements of the matrices of potential and capacitive coefficients is estimated as less than 0.1%.

Consider the propagation of electromagnetic energy through multiconductor three-phase high-voltage transmission line with arbitrary number of conductors. The mathematical formulation of the problem represents the system of partial differential equations known as transmission line equations. When constructing the capacitance matrix of an electric line, it is necessary to solve the boundary problems for the Laplace equation with nonlocal boundary conditions. The aim of this research is to develop a new method for constructing a potential coefficients matrix, which allows us to express the voltage vector in the wires through the charge vector. The application of this method leads to the necessity of solving a boundary value problem for Laplace equations with nonlocal boundary conditions. A new type of non-local boundary conditions has been identified, which has not been previously investigated. The boundary conditions of this type are represented by the contour integral over the boundary of the region from an unknown potential. In the general case, the obtained problem does not have a unique solution. The theorem of existence and uniqueness of the solution for the boundary value problem for the Laplace equation with nonlocal boundary conditions is proved. However, the requirement that the potential on the wire surface is constant allows us to prove the existence and uniqueness of the solution for considered problem. In order to illustrate the application of the developed approach we have solved numerically the problem for the sector-shaped cable with three cores. The values of potential and capacitive coefficients obtained by calculation are given. The error of the values of the diagonal elements of the matrices of potential and capacitive coefficients is estimated as less than 0.1%.