Analysis of eddy currents by the method of generalized Fourier expansion and by the boundary element method coupled with impedance boundary conditions in axisymmetric problems

The effective utilization or reduction of eddy current losses is of great significance for the electrical devices in the industrial applications, for instance, electric machine, transformer, electromagnetic induction heating, electromagnetic forging, wireless power transfer, as well as non-destructive detection. As a consequence, the prediction and evaluation of eddy currents in such electrical devices is fundamental and of highly desirable in the stages of design. As known, the eddy current analysis is more complicated compared with the evaluations of the electrostatic field and the steady magnetic field. Specifically, the efficient and accurate computation of eddy currents has been widely emphasized in the field of electromagnetism. Typically, the eddy current analysis can be implemented by means of analytical or numerical solutions. Generally, the analytical approaches provide an immense advantage that the influence of a geometrical or a material parameter on the eddy current distribution can be easily estimated. However, analytical solutions only suitable and applicable for very few eddy current problems as a matter of fact with simple geometries. On the contrary, for eddy current problems with slightly more geometrical complexities, numerical approaches, as supplementary tools must be applied. Remark that a critical drawback of numerical solutions is that the geometrical and material parameters of an electromagnetic problem are all implicitly involved in the numerical results. In order to fully evaluate the contributions of various of parameters on eddy current distribution, the numerical solutions must be carried out by setting the parameters with a set of prescribed values, which inevitably leads to an expensive computation burden. In recent years, the numerical solution based on the boundary element method (BEM) coupling with high-order surface impedance boundary conditions (SIBC) has been raised considerable concern. Compared with the standard finite element method (FEM), the BEM-SIBC holds primary merits in terms of a reduced computational dimensions in spatial and suitability for large-scale and open domain electromagnetic problems. Furthermore, the BEM-SIBC shows higher computational efficiency, especially for the case coupling with the classic perturbation theory (PT). Interestingly that the solutions in the entire valid frequency range can be obtained by only once computation. For the two-dimensional axisymmetric eddy current problems, this dissertation conducts a systematic and in-depth investigation on the analytical approach and BEM-SIBC with specific emphases on the applied theories and implementation techniques. The main research activities and highlighted achievements are summarized into four aspects and are structured as follows: (1) By utilizing the method of generalized Fourier expansion, an analytical solution for axisymmetric eddy current problems with multi-layer media is proposed. It is applicable for the cases involving multiple series connected rectangular cross-sectional windings. Specifically, the integral representations of the magnetic vector potential in multi-media, the induced voltage of multiple series connected rectangular cross-sectional windings, and the equivalent impedance of the entire windings are formulated. Remarkably, the computational accuracy of the equivalent impedance is considerably improved compared with a conventional analytical solution. For the ease of the design and transient analysis of induction heating systems, a broadband circuit model for the equivalent impedance is implemented with the aid of the vector fitting technique and a circuit synthesis approach. Good agreement for the impedance is achieved between the simulation of the equivalent circuit model and the analytical solution. (2) A numerical solution coupling the BEM with SIBC (denoted by BEM-SIBC) for axisymmetric eddy current problems is proposed based on the magnetic scalar potential. On the basis of the interface conditions of the adjacent media, the boundary element integral equation coupling with SIBC is derived. Subsequently, the PT is adopted to approximate the magnetic scalar potential and magnetic field intensity. It is implemented with the aid of a small parameter which actually is selected to be the product of the relative permeability and the skin depth in conductive regions. Afterwards, transferring the approximate expansions into dimensionless quantities and then substituting them into the diffusion equation to yield the SIBCs with zero order, the 1st order, the 2nd order, and the 3rd order successively. In terms of the numerical implementation of BEM-SIBCs, the quadratic isoparametric element is applied for the discretization and the adaptive integral technique is employed for the computation of coefficient matrices of BEM. In addition, the cubic smoothing spline method is adopted to approximate the high-order derivatives involved in BEM-SIBCs. Numerical validations indicate that comparing with FEM, a significant improvement of computational efficiency and reduction of memory requirement is obtained, especially when a frequency sweep analysis is carried out. It is worth mentioning that BEM coupling with the 3rd SIBC shows the highest accuracy in the entire frequency range of validity. (3) Based on the proposed BEM-SIBC technique, an energy approach for the extraction of equivalent internal impedance for axisymmetric eddy current applications is proposed. A small parameter is predefined and selected to be the product of the relative permeability and the skin depth in conductive regions. Then the electric field is approximated by a power series with respect to the chosen small parameter. Subsequently, in conductive regions, the energy and the equivalent internal impedance can be evaluated by applying the Poynting’s theorem. The highest accuracy can be found from the BEM coupling with the 3rd SIBC, taking a flat spiral air-core inductor as a typical example. Compared with FEM, the proposed BEM-SIBC assures a significant reduction of the computational burden. (4) A field splitting technique is proposed based on the BEM-SIBC approach for the solution of axisymmetric eddy current problems. In particular, the electromagnetic field is split into two components, that is, a source component and an eddy component. Subsequently, a boundary integral equation coupling with the SIBC is formulated by substituting the source and eddy components. Specifically, the source component can be determined by using BEM-SIBC and the Ampere’s law. As a further step, in the conductive regions, the induced voltage, and the total equivalent impedance of the entire system can be evaluated based on the pre-computed source component. Remarkably, compared with the BEM-SIBC based on the magnetic vector potential, the presented boundary integral expression involving the source and eddy components reveals to be a concise representation and a significant reduction in terms of the BEM programming complexity. Three dedicated test cases, that is, two typical spiral air-core inductors and a coaxial toroidal magnetic coupling wireless power transfer system used in electric vehicles are carried out. A high computational accuracy is obtained, and an efficient computation is achieved under a frequency sweep analysis.

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