There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Introduction to Electromagnetic Compatibility

The finite-element method (FEM) is a full-wave numerical method that discretizes the variational of a functional. The evolution of this method within the scope of electromagnetics traces back to the solving of two classes of problems, namely, the eigenmode problems and the deterministic problems. If we try to use some examples to illustrate the most typical and the most complete techniques with the most complete solution, the eigenmode problem of a dielectrically loaded waveguide and the wave propagation in a three-dimensional (3D) discontinuous waveguide are good candidates representing the eigenmode and the closed-domain solutions, respectively. As for the open-domain scattering problem and radiating problems, the authors consider the essential and key parts are presented in solving the 3D scattering problems. For this reason, we will illustrate the theoretical basics, the critical solving techniques and the typical skills involved in FEM through solving of the above three specific problems. At the end of this chapter, we will also briefly review the FEM solution for some other problems.

Finite-Element Method

Various magnetic vector potential formulations for the eddy-current problem are reviewed. The uniqueness of the vector potential is given special attention. The aim is to develop a numerically stable finite-element scheme that performs well at low and high frequencies, does not require an unduly high number of degrees of freedom, and is capable of treating multiple connected conductors. >

On the use of the magnetic vector potential in the finite-element analysis of three-dimensional eddy currents

Numerical Methods for Scientists and Engineers

One way to deal with non‐linearity in the case of time harmonic excitation of the electric and magnetic fields in ferromagnetic media is to introduce an effective material to model the ferromagnetic region. This fictitious material is constant throughout a period but inhomogeneous and it takes into account the non‐linear relationship between the field quantities. Several methods to create an effective magnetization curve are presented and different finite element formulations are applied to these. Some of these methods are known from the literature for 1D and 2D problems but not for 3D ones and they have only be used for a magnetic vector potential formulation so far. In the present paper they are extended for general vector and scalar potential formulations. Further possible ways to introduce such an effective material to take into account saturation effects are shown. All these different methods are investigated on a non‐linear 3D time harmonic eddy current problem using complex formalism.

Time harmonic eddy currents in non‐linear media

The geometric multigrid (MG) method using the point-wise iterative method as the smoother converges slowly for 3-D magnetic fields calculation when there are thin edge elements in the finite element mesh. This paper proposes an edge based plane smoother so that the convergence of the MG method can be significantly improved. Numerical examples show that the MG method with the proposed smoother retains its good efficiency even for problems discretized by edge elements with extremely high aspect ratios.

Geometric Multigrid With Plane Smoothing for Thin Elements in 3-D Magnetic Fields Calculation

A method based on variational principles has been developed for the computation of the eddy current distribution and hence the driving torque of single-phase kilowatt-hour meters. The airgap field due to the voltage and current coils in the absence of the disk is presumed to be given, and the resultant air-gap field, as well as the eddy currents induced by it, are determined. The reaction field of the eddy currents is taken into account approximately. Global approximation is employed to yield the field quantities as expansions in terms of suitable analytical functions. A comparison is made between computed and measured torques.

Calculation of the eddy current distribution in the disk of single-phase kilowatt-hour meter by variational method

This paper presents an extended finite element based model approach for permanent magnet synchronous machines considering dynamic eccentricity effects of the rotor. This includes additional displacement forces acting on rotor and stator as well as the influence of rotor eccentricity to the machine torque and the back electromotive force. Furthermore, the work flow for this model is shown.

An extended finite element based model approach for permanent magnet synchronous machines including rotor eccentricity

A method for steady-state calculation of mechanical or structural systems formulated by the finite element method (FEM) is presented. The procedure introduced is used to calculate the mechanical vibrations caused by the electromagnetic excitation of induction motors. Compared to other approaches, this method enables the free choice of the time discretisation algorithm. The method is presented for second order systems and demonstrated by a numerical example.

Oszkar Biro

Papers

Time harmonic eddy currents in non‐linear media

Geometric Multigrid With Plane Smoothing for Thin Elements in 3-D Magnetic Fields Calculation

Calculation of the eddy current distribution in the disk of single-phase kilowatt-hour meter by variational method

An extended finite element based model approach for permanent magnet synchronous machines including rotor eccentricity

Direct steady-state computation of mechanical vibrations in electrical machines