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Purpose To understand the behavior of the magnetization processes in ferromagnetic materials in function of temperature, a temperature-dependent hysteresis model is necessary. This study aims to investigate how temperature can be accounted for in the energy-based hysteresis model, via an appropriate parameter identification and interpolation procedure. Design/methodology/approach The hysteresis model used for simulating the material response is energy-consistent and relies on thermodynamic principles. The material parameters have been identified by unidirectional alternating measurements, and the model has been tested for both simple and complex excitation waveforms. Measurements and simulations have been performed on a soft ferrite toroidal sample characterized in a wide temperature range. Findings The analysis shows that the model is able to represent accurately arbitrary excitation waveforms in function of temperature. The identification method used to determine the model parameters has proven its robustness: starting from simple excitation waveforms, the complex ones can be simulated precisely. Research limitations/implications As parameters vary depending on temperature, a new parameter variation law in function of temperature has been proposed. Practical implications A complete static hysteresis model able to take the temperature into account is now available. The identification is quite simple and requires very few measurements at different temperatures. Originality/value The results suggest that it is possible to predict magnetization curves within the measured range, starting from a reduced set of measured data.

Purpose This paper aims to develop a methodology for progressive finite element (FE) modeling of transformers, from simple to complex models of both magnetic cores and windings. Design/methodology/approach The progressive modeling of transformers is performed via a subproblem (SP) FE method. A complete problem is split into SPs with different adapted overlapping meshes. Model refinements are performed from ideal to real flux tubes, one-dimensional to two-dimensional to three-dimensional models, linear to nonlinear materials, perfect to real materials, single wire to volume conductor windings and homogenized to fine models of cores and coils, with any coupling of these changes. Findings The proposed unified procedure efficiently feeds each SP via interface conditions (ICs), which lightens mesh-to-mesh sources transfers and quantifies the gain given by each refinement on both local fields and global quantities, with a clear view on its significance to justify its usefulness, if any. It can also help in education with a progressive understanding of the various aspects of transformer designs. Originality/value Models of different accuracy levels are sequenced with successive additive corrections supported by different adapted meshes. The way the sources act at each correction step, up to the full models with their actual geometries, is given a particular care and generalized, allowing the proposed unified procedure. For all the considered corrections, the sources are always of IC type, thus only needed in layers of FE along boundaries, which lightens the required mesh-to-mesh projections between subproblems.

The electromagnetic interference between electronic systems or between their components influences the overall performance. It is important thus to model these interferences in order to optimize the position of the components of an electronic system. In this paper, a methodology to construct the equivalent model of magnetic field sources is proposed. It is based on the multipole expansion, and it represents the radiated emission of generic structures in a spherical reference frame. Experimental results for different kinds of sources are presented illustrating our method.

Purpose - This paper seeks to develop a sub-domain perturbation technique to efficiently calculate strong skin and proximity effects in conductors within frequency and time domain finite element (FE) analyses.Design methodology approach - A reference eddy current FE problem is first solved by considering perfect conductors. This is done via appropriate boundary conditions (BCs) on the conductors. Next the solution of the reference problem gives the source for eddy current FE perturbation sub-problems in each conductor then considered with a finite conductivity. Each of these problems requires an appropriate volume mesh of the associated conductor and its surrounding region.Findings - The skin and proximity effects in both active and passive conductors can be accurately determined in a wide frequency range, allowing for precise losses calculations in inductors as well as in external conducting pieces.Originality value - The developed method allows one to accurately determine the current density distributions and ensuing losses in conductors of any shape, not only in the frequency domain but also in the time domain. Therefore, it extends the domain of validity and applicability of impedance-type BC techniques. It also offers an original way to uncouple FE regions that allows the solution process to be lightened, as well as efficient parameterized analyses on the signal form and the conductor characteristics.

Purpose - To develop a homogenization technique to directly and efficiently take the eddy current effects in laminated magnetic cores within time domain finite element (FE) analyses.Design methodology approach - The technique is developed for being used within a 3D magnetodynamic b-conform FE formulation, e.g. using a magnetic vector potential. To avoid a fine FE discretization of all the laminations of a magnetic core, this one is considered as a source region that carries predefined current and magnetic flux density distributions describing the eddy currents and skin effect along each lamination thickness. Both these distributions are related and are first approximated with sub-basis functions. Through the homogenization or averaging of the sub-basis functions contributions in the FE formulation, the stacked laminations are then converted into continua, thus implicitly considering the eddy current loops produced by parallel magnetic fluxes. The continuum is then approximated with classical FE basis functions and can be defined on a coarser discretization.Findings - The developed method appears attractive for directly and efficiently taking into account within finite element analyses the eddy current effects, i.e. the associated losses and magnetic flux reduction, that are particularly significant for high frequency excitations. The time domain analysis allows the consideration of both non-linear and transient phenomena.Originality value - The averaging of sub-basis functions contributions, describing fine distributions of fields, in an FE formulation leads to an original way of homogenizing laminated regions. The proposed method is naturally adapted for time domain analyses and in some sense generalizes what can be done more easily in the frequency domain.

Purpose – The purpose of this paper is to develop a subproblem finite element method for progressive modeling of lamination stacks in magnetic cores, from homogenized solutions up to accurate eddy current distributions and losses. Design/methodology/approach – The homogenization of lamination stacks, subject to both longitudinal and transversal magnetic fluxes, is first performed and is followed by local correction subproblems in certain laminations separately, surrounded by their insulating layers and the remaining laminations kept homogenized. The sources for the local corrections are originally defined via interface conditions to allow the coupling between homogenized and non-homogenized portions. Findings – The errors proper to the homogenization model, which neglects fringing effects, can be locally corrected in some selected portions via local eddy current subproblems considering the actual geometries and properties of the related laminations. The fineness of the mesh can thus be concentrated in these portions, while keeping a coupling with the rest of the laminations kept homogenized. Research limitations/implications – The method has been tested on a 2D case having linear material properties. It is however directly applicable in 3D. Its extension to the time domain with non-linear properties will be done. Originality/value – The resulting subproblem method allows accurate and efficient calculations of eddy current losses in lamination stacks, which is generally unfeasible for real applications with a single problem approach. The accuracy and efficiency are obtained thanks to a proper refined mesh for each subproblem and the reuse of previous solutions to be locally corrected only acting in interface conditions. Corrections are progressively obtained up to accurate eddy current distributions in the laminations, allowing to improve the resulting global quantities: the Joule losses in the laminations, and the resistances and inductances of the surrounding windings.

Purpose – The purpose of this paper is to introduce a novel methodology for uncertainty quantification in large-scale systems. It is a non-intrusive approach based on the unscented transform (UT) but it requires far less simulations from a EM solver for certain models. Design/methodology/approach – The methodology of uncertainty propagation is carried out adaptively instead of considering all input variables. First, a ranking of input variables is determined and after a classical UT is applied successively considering each time one more input variable. The convergence is reached once the most important variables were considered. Findings – The adaptive UT can be an efficient alternative of uncertainty propagation for large dimensional systems. Originality/value – The classical UT is unfeasible for large-scale systems. This paper presents one new possibility to use this stochastic collocation method for systems with large number of input dimensions.

Purpose – The purpose of this paper is to develop a subproblem method (SPM) for progressive modeling of inductors, with model refinements of both source conductors and magnetic cores. Design/methodology/approach – The modeling of inductors is split into a sequence of progressive finite element (FE) SPs. The source fields (SFs) generated by the source conductors alone are calculated at first via either the Biot-Savart (BS) law or FEs. With a novel general way to define the SFs via interface conditions (ICs), to lighten their evaluation process, the associated reaction fields for each added or modified region, mainly the magnetic cores, and in return for the source conductor regions themselves when massive, are then calculated with FE models. Changes of magnetic regions go from perfect magnetic properties up to volume linear and nonlinear properties, and from statics to dynamics. Findings – For any added or modified region, the novel proposed ICs to define the SFs appear of general usefulness, which opens the method to a wide range of model improvements. Originality/value – The resulting SPM allows efficient solving of parameterized analyses thanks to a proper mesh for each SP and the reuse of previous solutions to be locally corrected, in association with novel SF ICs that strongly lighten the quantity of BS evaluations. Significant corrections are progressively obtained for the fields, up to nonlinear magnetic core properties and skin and proximity effects in conductors, and for the related inductances and resistances.

In this paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator.

Purpose - The purpose of this paper is to develop a sub-domain perturbation technique for refining magnetic circuit models with finite element (FE) models of different dimensions.Design methodology approach - A simplified problem considering ideal flux tubes is first solved, as either a 1D magnetic circuit or a simplified FE problem. Its solution is then corrected via FE perturbation problems considering the actual flux tube geometry and the exterior regions, that allow first 2D and then 3D leakage fluxes. Each of these sub-problems requires an appropriate proper volume mesh, with no need of interconnection. The solutions are transferred from one problem to the other through projections of source fields between meshes.Findings - The developed perturbation FE method allows to split magnetic circuit analyses into subproblems of lower complexity with regard to meshing operations and computational aspects. A natural progression from simple to more elaborate models, from 1D to 3D geometries, is thus possible, while quantifying the gain given by each model refinement and justifying its utility.Originality value - Approximate problems with ideal flux tubes are accurately corrected when accounting for leakage fluxes via surface sources of perturbations. The constraints involved in the subproblems are carefully defined in the resulting FE formulations, respecting their inherent strong and weak nature. As a result, an efficient and accurate computation of local fields and global quantities, i.e. flux, MMF, reluctance, is obtained. The method is naturally adapted to parameterized analyses on geometrical and material data.