Identification of the continuous pinning field probability density 

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Validation with 2D FE Simulations

To test the accuracy of hysteresis models in FEA (Finite Element Analysis), simulation results can be compared to measurements on a test-case for the validation of magnetic field analysis. This chapter applies the EB hysteresis model to the FE simulation of a real transformer application [118] designed in the Ampère laboratory. This represents a test-case, inspired by [119], to benchmark the predictive capability of the hysteresis model, considering different current excitations of the transformer winding system, which are adapted to yield sinusoidal, phase-shifted, distorted or rotating magnetic flux density waveforms. Measured data are compared to the simulated ones for various excitation cases.

Experimental setup

The setup has been designed and built especially for this purpose. The geometry of the transformer is shown in Figure 4.1, for which the parameters are listed in Table 4.1. The D.U.T consists of a pair of E-cores made of the same material, which forms a closed magnetic flux path (cf. Figure 4.2). The material is a very common material (soft ferrite Mn-Zn 3C90 from Ferroxcube [103], previously characterised) used in power electronics.
Two windings of 15 turns are placed on the external limbs. The windings can be connected in series or supplied by two independent controlled voltage sources. Unidirectional or rotating fields in some parts are then possible. For each configuration, both the applied voltage and the currents through the windings are measured. With the help of suitable supports built by means of a 3D printer, the sensor coils C1 and C2, consisting of 20 turns each, are used to estimate the flux going in the central leg from each winding.
This provides a test-case for the validation of a 2D magnetic field analysis as well as vector hysteresis and presence of multiple harmonic signals. Several test cases are defined depending on the supplied conditions of the windings (cf. Figure 4.3), with various maximal amplitude field value components vp, and phase shifting angles . In that way, a set of different excitation source fields is built, made of two different amplitudes, vp = 1 V (LA = low amplitude) or 5 V (HA = high amplitude), associated with two values, 0° (unidirectional case) and 90° (rotating case). The test-case has different configurations, but attention will be paid to following ones:
• Case 1 is tested to validate unidirectional waveforms.
• Case 2 is appropriate for checking vector hysteresis modelling, with rotational flux.
• Case 3 is convenient for checking distorted multi-harmonic signals under unidirectional or rotating fluxes.

FE simulations

This section introduces the simulations built in the open source Gmsh/GetDP FE code [120] by means of the 2D FE magnetodynamic a −v formulation (Appendix B), in which the EB model (simplified differential approach) has been implemented [47]. Indeed, as ferrite is an insulator, and coils are modelled as stranded inductors, there are no eddy currents, hence in practice the formulation reduces to the simpler magnetostatic a−formulation. This has been previously elaborated and tested on different cases of increasing geometrical complexity, such as the square case and the TEAM 32 transformer [47]- [120].
The windings are considered as stranded inductors and each of them is represented in the cross-section by two rectangle regions around a core leg. The electric global quantities are used in the definition of voltage sources and for the coupling with circuit equations. A coupling with external circuits in the FE software enables the imposition of voltage-driven sources. The test-case transformer can be considered as a 2D case, in which the fields h and b lie in the xy-plane and the current density j is assumed perpendicular to the domain. Thanks to the 2D limitation, the a − v formulation is expressed in terms of a potential vector a, where the component along the z-axis, az, represents the unknown of the problem.
Moreover, due to symmetry, only half of the transformer cross-section can be onsidered, as seen in Figure 4.4. The geometry is discretized using a mesh made of 4420 triangles with 2000 nodes. Points 1-2-3-4 have been fixed in order to calculate the vector potentials az(1), az(2), az(3) and az(4). The aim is to compute the magnetic flux distributions on the core region, in particular the flux going in the central leg from each external winding.
As concerns the model parameters choice, simulations are performed adopting N = 4 cells for the EB model, with the previously identified discrete parameters (!k, k) for T = 293 K (cf. Table 9.1). The anhysteretic saturation law is firstly approximated by a hyperbolic tangent function such as (2.18) with saturation magnetic polarisation Ja = 0.38 T and ha = 53 A/m, a parameter affecting the initial slope of the curve.

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Table of contents :

I English version 
1 State of the art 
1.1 Origins of Magnetism
1.1.1 Basic concepts
1.1.2 Different types of behaviour in magnetic materials
1.2 Ferromagnetism: from mesoscopic to macroscopic scale
1.2.1 Mesoscopic scale
1.2.2 Macroscopic scale: the hysteresis cycle
1.3 Classification of ferromagnetic materials
1.4 Iron loss modelling
1.4.1 Empirical approaches
1.4.2 Bertotti’s loss decomposition
1.5 Hysteresis models
1.5.1 Static hysteresis models
1.5.1.1 The Preisach model
1.5.1.2 The Jiles-Atherton model
1.5.1.3 Henrotte’s Energy-Based model
1.5.1.4 Static hysteresis and temperature
1.5.2 Dynamic hysteresis models
1.5.2.1 Dynamical with Static Feedback model
1.5.2.2 Dynamical with Wall Motion model
1.5.2.3 Loss Surface model
1.5.3 Dynamic-thermal problematics
2 Magnetic Measurements 
2.1 Characterisation by ring method
2.2 Quasi-static measurements
2.2.1 Temperature-dependent measurements
2.2.1.1 Simple excitation waveforms
2.2.1.2 Complex excitation waveforms
2.2.2 Measurement results
2.3 High Frequency (HF) measurements
2.4 Identification of the Energy-Based model
2.4.1 Identification of the continuous pinning field probability density
2.4.2 Discretization of the model
3 Static modelling 
3.1 Effect of temperature on magnetic properties
3.2 Parameters identification in function of temperature
3.3 Validation of temperature-dependent hysteresis
3.4 EB model: accuracy and robustness analyses
3.5 Validation on complex waveforms
3.6 A first assessment
3.7 Proposed temperature extension
3.7.1 Continuous !(|T) curves interpolation
3.7.2 Discrete (!k, k) sets function of temperature
3.8 Conclusion
4 Validation with 2D FE Simulations 
4.1 Experimental setup
4.2 FE simulations
4.2.1 Case 1 (series)
4.2.2 Case 2 (rotating fields)
4.2.3 Case 3 (multi-harmonics)
4.3 Discussion
4.4 Conclusion
5 Conclusion and perspectives 
5.1 Contributions of this work
5.2 Perspectives
5.2.1 Dynamic modelling
5.2.2 Identification protocols
5.2.3 Link with microstructure
II Version française 
6 État de l’art 
6.1 Origines du magnétisme
6.2 Classification de matériaux selon leurs propriétés magnétiques
6.2.1 Ferromagnétisme à différentes échelles
6.3 Vue d’ensemble des modèles de pertes fer
6.3.1 Modèle de Steinmetz
6.3.2 Modèle de Bertotti
6.4 Les modèles d’hystérésis
7 Moyens de mesure et caractérisation expérimentale
7.1 Mesures quasi-statiques en fonction de la température
7.1.1 Formes d’ondes d’excitation simples
7.1.2 Formes d’ondes d’excitation complexes
7.2 Résultats
7.3 Mèthode d’identification des paramètres
8 Modélisation statique
8.1 Effet de la température sur les propriétés magnétiques
8.2 Identification des paramètres en fonction de la température
8.3 Validation de mesures « standards »
8.4 Modèle EB : analyses de précision et de robustesse
8.5 Validation des mesures « complexes »
8.6 Extension du modèle : prise en compte des variations de température
8.6.1 Interpolation des courbes !(|T) continues
8.6.2 Paramètres discrets (!k, k) en fonction de la température
8.7 Conclusion
9 Validation avec simulations aux éléments finis 
9.1 Dispositif expérimental
9.2 Simulations aux éléments finis
9.2.1 Cas 1 (série)
9.2.2 Cas 2 (déphasage entre les sources de tension)
9.2.3 Cas 3 (signaux multi-harmoniques)
9.3 Discussion
10 Conclusions et perspectives

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