Full–wave time–harmonic electromagnetic model of thin–wire conductor in a multilayer conductive medium

Ivan Krolo
January 2021
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Abstract

In this doctoral dissertation, a 3D exact harmonic electromagnetic model of an infinitesimal thin-wire conductor (ITV) in a horizontally layered multilayer conductive medium, consisting of air and multilayer soil, has been developed. Based on this 3D exact electromagnetic model, a 2D exact harmonic electromagnetic model of an infinitely long insulated thin-wire conductor (BDITV) has been developed, as well as a 2D exact harmonic electromagnetic model based on it for calculating the self and mutual longitudinal unit impedances of rectilinear mutually parallel BDITVs in a horizontally layered multilayer conductive medium, consisting of air and multilayer soil. The 3D exact harmonic electromagnetic model of the ITV contains numerically demanding 3D Sommerfeld integrals, while the 2D exact harmonic electromagnetic model for calculating the self and mutual longitudinal unit impedances of rectilinear mutually parallel BDITVs contains equally numerically demanding 2D Sommerfeld integrals. In the theoretical development of the 3D exact harmonic electromagnetic model of the ITV, in order to more easily satisfy the boundary conditions, the arbitrarily oriented ITV is decomposed into two components: a vertical infinitesimal thin-wire conductor (VITV) and a horizontal infinitesimal thin-wire conductor (HITV). From the general solutions of the Helmholtz equations for the components of the vector magnetic potential of the VITV and HITV and the boundary conditions, systems of linear equations were obtained, in which the unknowns are the kernels of the spectral Green’s functions of the VITV and HITV. By very demanding analytical solution of these systems of linear equations, analytical expressions for the kernels of spectral Green’s functions of the vector magnetic potential components of VITV and HITV were originally obtained, and from them, analytical expressions for the kernels of spectral Green’s functions of the scalar electric potential, electric field strength components and magnetic induction components of an arbitrarily oriented ITV in a horizontally layered multilayer conductive medium were obtained. The total number of soil layers, the layer in which the source is located and the layer in which the observation point is located are completely arbitrary. 3D Sommerfeld integrals connect the spatial and spectral Green’s functions of Lorentz potentials, electric field strength components and magnetic induction components of ITV. In the theoretical development of the 2D exact harmonic electromagnetic model of the BDITV parallel to the ground surface, the spatial Green’s function of the vector magnetic potential of the BDITV in a horizontally layered multilayer medium was obtained by integrating the spatial Green’s function of the horizontal component of the vector magnetic potential of the HITV along the BDITV axis. In this way, a 2D exact electromagnetic model of the BDITV was obtained from the 3D exact electromagnetic model of the HITV. In the electromagnetic model of the BDITV, instead of 3D Sommerfeld integrals, 2D Sommerfeld integrals appear, while the spectral Green’s functions of the vector magnetic potential of the BDITV and the horizontal components of the vector magnetic potential of the HITV are identical. From the originally developed 2D exact harmonic electromagnetic model for calculating the self and mutual longitudinal unit impedances of rectilinear mutually parallel BDITVs in a multilayer horizontally layered medium, with some neglects, the generally known Carson and Sunde formulas for calculating the self and mutual longitudinal unit impedances of rectilinear mutually parallel BDITVs in a two-layer horizontally layered medium, consisting of air and homogeneous soil, were obtained. In this doctoral dissertation, numerical algorithms for the numerical calculation of 3D and 2D Sommerfeld integrals were also originally developed, which are based on a combination of polynomial interpolation of spectral Green’s functions by parts of the integration curve and analytical integration. In addition, the existing Mosig-Michalski algorithm, which was developed for calculating 3D Sommerfeld integrals for a horizontally layered multilayer ideal dielectric, was modified and adapted to the peculiarities of 3D and 2D Sommerfeld integrals in the case of a horizontally layered multilayer conductive medium, consisting of air and multilayer soil. Numerous numerical tests have shown that both numerical algorithms efficiently and with high accuracy calculate the numerically demanding 3D and 2D Sommerfeld integrals for a wide range of variable parameters.

Type
Thesis
Full-Wave Time-Harmonic Electromagnetic Model Spectral-Domain Green's Functions Spatial Domain Green's Functions Infinitesimal Thin-Wire Conductor Infinitely Long Insulated Thin-Wire Conductor Series Per-Unit-Length Self-Impedance Series Per-Unit-Length Mutual Impedance Sommerfeld Integrals
Ivan Krolo
Ivan Krolo
Assistant Professor | Department of Theoretical Electrical Engineering and Modelling

Researcher and assistant professor at the Faculty of Electrical Engineering, Mechanical Engineering, and Naval Architecture in Split, with research focus on the development of numerical models for grounding systems and issues related to electrical safety in low-voltage and high-voltage systems.