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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.

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.

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.

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.

To develop a sub‐domain perturbation technique for efficiently modeling moving systems in magnetodynamics with a magnetic field h‐conform finite element (FE) formulation.

A reference problem is first solved in a global mesh excluding some moving regions and thus avoiding the inclusion of their meshes. Its solution gives the sources for a sequence of perturbation problems with the supplementary moving magnetic and conductive regions. Each of these sub‐problems requires an appropriate proper volume mesh of the associated moving region and its surrounding region, with no need of interconnection. The solutions are transferred from one problem to the other through projections of source fields between meshes.

The consideration of sub‐problems and associated sources, in a sequence of perturbation problems, leads to a significant speed‐up of the repetitive solutions in analyses of moving systems. A free movement in any direction can be considered with no need of remeshing.

When working with the perturbation fields, the volume sources can be limited to the moving regions, what allows for homogeneous perturbation boundary conditions and reduces the computational efforts for projecting and evaluating the sources. The curl‐conformity of the unknown magnetic field is preserved during the whole process thanks to the use of edge FEs for both the magnetic field and the intermediate source quantities. The sub‐problem approach also gives an easy way to directly express the time derivatives in moving frames.

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.

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.

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.

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.

The purpose of this paper is to deal with the correction of the inaccuracies near edges and corners arising from thin shell models by means of an iterative finite element subproblem method. Classical thin shell approximations of conducting and/or magnetic regions replace the thin regions with impedance-type transmission conditions across surfaces, which introduce errors in the computation of the field distribution and Joule losses near edges and corners.

In the proposed approach local corrections around edges and corners are coupled to the thin shell models in an iterative procedure (each subproblem being influenced by the others), allowing to combine the efficiency of the thin shell approach with the accuracy of the full modelling of edge and corner effects.

The method is based on a thin shell solution in a complete problem, where conductive thin regions have been extracted and replaced by surfaces but strongly neglect errors on computation of the field distribution and Joule losses near edges and corners.

This model is only limited to thin shell models by means of an iterative finite element subproblem method.

The developed method is considered to couple subproblems in two-way coupling correction, where each solution is influenced by all the others. This means that an iterative procedure between the subproblems must be required to obtain an accurate (convergence) solution that defines as a series of corrections.

Two complementary 3D finite element formulations, with either the magnetic field or the magnetic vector potential as unknowns, are developed to deal with the modeling of eddy currents in electrical steel laminations. The magnetic flux through the flux gates of the conducting region is imposed via the boundary terms of the weak formulations, in a natural way thanks to the use of edge finite elements. The two formulations are applied to a simple 1D eddy current problem with analytical solution. As a practical 3D application example, a T‐joint region of an electrical steel lamination is considered.

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.

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.

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.

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.

Develops a method to take the eddy currents in stacked thin regions, in particular lamination stacks, into account with the finite element method using the 3D magnetic vector potential magnetodynamic formulation. It consists in converting the stacked laminations into continuums with which terms are associated for considering the eddy current loops produced by both parallel and perpendicular fluxes. Non‐zero global currents can be considered in the laminations, in particular for studying the effect of imperfect insulation between their ends. The method is based on an analytical expression of eddy currents and is adapted to a wide frequency range.

Two sets of dual magnetodynamic and magnetostatic finite element formulations taking thin conducting magnetic shells into account are proposed. The abstraction of the thin region from the computational domain is performed by an appropriate treatment of the surface integral terms arising in the weak formulations. Results are presented for two three‐dimensional test‐problems.

Connection boundary conditions are studied with the finite element method using different types of mixed finite elements, i.e. nodal, edge and facet elements of different shapes and degrees, used in both b‐ and h‐conform formulations. The developed associated tools are first applied to periodicity boundary conditions before being applied to the treatment of the moving band in 2D and 3D. This step by step approach enables their validation before pointing out the effect of the considered elements on the accuracy of the moving band method. A special attention is given to the consistency of these boundary conditions with gauge conditions and source magnetic fields.

This paper deals with the two‐dimensional finite element analysis in the frequency domain of saturated electromagnetic devices coupled to electrical circuits comprising nonlinear resistive and inductive components. The resulting system of nonlinear algebraic equations is solved straightforwardly by means of the Newton‐Raphson method. As an application example we consider a three‐phase transformer feeding a nonlinear RL load through a six‐pulse diode rectifier. The harmonic balance results are compared to those obtained with time‐stepping and the computational cost is briefly discussed.