The hemispherical ground electrode is a basic electrode whose analysis appears in many textbooks on electromagnetics in chapters dedicated to steady currents. Considering a soil with a given ...resistivity and an electrode with a given perimeter, the electrode DC resistance is simply calculated from the ratio resistivity/perimeter. Strangely, the generalization of this result to AC regimes is missing. The issue of the frequency-domain impedance of the hemispherical ground electrode has been avoided in the literature despite its trivial geometry. But the problem is indeed not easy; electromagnetic field calculation involves Legendre and Bessel functions; the application of boundary conditions involves an infinite set of points, and some integrals involved need to be calculated recursively. We analyzed the math and physics of the problem but failed to find a closed-form solution. This article with “negative results” can, however, be useful; on one hand it may prevent researchers from wasting their time following the same steps, and, on the other hand, it may attract the interest of new researchers to the subject, ultimately, accelerating its analytical solution (if the solution exists).
This article is the authors’ last contribution to a trilogy of research papers submitted to Energies’ Special Issue on Electromagnetic Field Computation, aimed at the theoretical analysis and ...numerical computation of the frequency-dependent complex impedance of hemispherical electrodes. In this work, we consider a pair of distant identical hemispherical electrodes buried in the ground, whose constitutive parameters (conductivity and permittivity) are assigned diverse values. Simulation experiments carried out using a full-wave finite element method, considering different combinations of the earth’s constitutive parameters, reveal that the grounding impedance of the electrode system can exhibit surprisingly varied frequency behavior. For frequencies close to zero, the impedance can start out inductive or capacitive, then go through a number of resonant transitions between inductive and capacitive states, finally tending towards purely resistive behavior. The results are interpreted using theoretical approximations valid for low- and high-frequency regimes.
Metallic electrodes are widely used in many applications, the analysis of their frequency-domain behavior is an important subject, particularly in applications related to earthing/grounding systems, ...from dc up into the MHz range. In this paper, a numerical evaluation of the frequency-dependent complex impedance of the hemispherical ground electrode is implemented. A closed-form solution for non-zero frequencies is still a difficult task to achieve as evidenced in a previous paper dedicated to the subject and, therefore, numerical approaches should be an alternative option. The aim of this article is to present a solution based on a numerical method using finite element analysis. In typical commercial FE tools, electric currents exhibit azimuthal orientation and, as such, the magnetic field has a null azimuthal component but non-null axial and radial components. On the contrary, a dual problem is considered in this work, with a purely azimuthal magnetic field. To overcome the difficulty of directly using a commercial FE tool, a novel formulation is developed. An innovative 2D formulation, the ι-form, is developed as a modification of the H-formulation applied to axisymmetric magnetic field problems. The results are validated using a classical 3D H-formulation; comparisons showed very good agreement. The electrode complex impedance is analyzed considering two different cases. Firstly, the grounding system is constituted by a hemispherical electrode surrounded by a remote concentric electrode; in the second case, the grounding system is constituted by two identical thin hemispherical electrodes. Computed results are presented and discussed, showing how the grounding impedance depends on the frequency and, also, on the radius of the remote concentric electrode (first case) or on the distance between the two hemispherical electrodes (second case).
In this paper, a magnetic field analysis is done for spherical shell-like induction machines with a zenithal progressive magnetic traveling wave, using an analytical model in spherical coordinates. ...These analytical models have major importance in the simulation, project, and preliminary optimization of devices due to its analytical solutions and physical insight. The magnetic field analysis is made for a shell-like induction machine for the study of its electromagnetic and mechanical quantities, as the magnetic field, induced currents, and electromagnetic torque. The developed model is validated, first with a set of generic windings distributions using the results of a commercial finite element analysis (FEA) tool and, second, validated by comparing with the results from the shell-like induction machine prototype. The results obtained from the analytical model are very close to the ones obtained by the FEA tool and from the prototype tests.
This paper is a contribution to the evaluation of electromagnetic fields due to underground power cables. Open boundary 2-D problems may be treated combining the finite element method (FEM) inside a ...region surrounded by a regular (cylindrical) fictitious surface with a semi-analytical boundary element method (BEM) to describe the field outside this region. For the outside region, a generalization of Pollaczek's solution is applied, taking into account eddy currents in the soil and the proximity effect between the soil/air plane surface and the cylinder fictitious surface with finite radius. The method is validated by using typical underground power cables where analytical methods may be applied accurately.
Medium frequency transmission line (MF TL) communications in circular cylindrical underground tunnels critically depend on the soil return impedance. The soil impedance can be approximately ...calculated using a very simple closed-form expression based on an image method-the so-called Dubanton approach. In this paper, we present an analytical exact method, based on electromagnetic field analysis, using Bessel and Hankel functions to correctly evaluate the soil impedance and the wave propagation parameters (attenuation, phase velocity, and characteristic impedance) that characterize the fundamental quasi-TEM mode propagating along circular tunnels. By comparing the computation results of both methods we have been able to quantify the errors involved in the Dubanton approach. The approximation developed by Wait is also used for verification purposes.
The design of magnetic cores can be carried out by taking into account the optimization of different parameters in accordance with the application requirements. Considering the specifications of the ...fast field cycling nuclear magnetic resonance (FFC-NMR) technique, the magnetic flux density distribution, at the sample insertion volume, is one of the core parameters that needs to be evaluated. Recently, it has been shown that the FFC-NMR magnets can be built on the basis of solenoid coils with ferromagnetic cores. Since this type of apparatus requires magnets with high magnetic flux density uniformity, a new type of magnet using a ferromagnetic core, copper coils, and superconducting blocks was designed with improved magnetic flux density distribution. In this paper, the designing aspects of the magnet are described and discussed with emphasis on the improvement of the magnetic flux density homogeneity (ΔB/B 0 ) in the air gap. The magnetic flux density distribution is analyzed based on 3-D simulations and NMR experimental results.
The main goal of this paper is the discussion of the energy transfer mechanisms between two concentric waveguides. An analytical model is developed for the electromagnetic propagation in two coupled ...concentric semiconductor waveguides. This model includes the calculation of the semiconductor refractive index taking into account the carrier densities, temperature and frequency. The model allows the analysis of the mechanisms of energy transfer between the two waveguides. As expected two types of energy loss are present, one related to the curvature of the waveguides and the other related to the absorption of the radiation in the semiconductor. Finally, the analyzed structure was optimized with regards to the above parameters.
This paper is a contribution to the analysis of magnetic field mitigation shielding of underground power cables. A hybrid finite-element method (FEM)/semi-analytical expansion is adopted for the ...numerical treatment of the open boundary magnetic field problem. Different constitutive materials for the magnetic shielding are studied, and comparison is established between conductive (aluminum) and ferromagnetic (steel) cases. A horizontal flat configuration for the underground cable is considered. It is concluded that the aluminum shielding is more efficient as far as magnetic field mitigation is concerned.
In this article, a two-dimensional hybrid finite-element formulation is proposed to further reduce the computational time required for the time simulation of superconducting (SC) bulks. Low ...computational time formulations are important for the design stage of electrical machines with superconductors, where optimization tools are typically used. The proposed hybrid formulation is based on the partitioned E - H electromagnetic scheme ( EHS ) formulation and the magnetic vector potential formulation ( A -formulation). The EHS formulation is used to solve the domains with highly nonlinear electric materials, while the A -formulation is used to solve the nonelectric domains. The main characteristics of the EHS are its partitioned formulation, where the matrix form of Ampère's and Faraday's laws is separated and computed a priori , and the electric resistivity is defined at each node. The EHS formulation is first compared with the H -formulation for a single SC domain and then integrated with an A -formulation and compared with the H and H - φ formulations to simulate a bulk surrounded by air. Results verify the excellent accuracy of the EHS and A-EHS formulations, and a promising reduction of computational time is observed, with more emphasis on low levels of magnetization of the SC sample. When compared with the H - φ formulation, a time reduction between 48% (6 T) and 90% (1 T) is obtained.
