Effect of Stress on Eddy Current Losses in Soft Magnetic Composites 

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EC Loss Density of Single Inclusion

If a magnetic ellipsoidal inclusion is placed in an infinite free space with initially uniform magnetic field, the magnetic field inside the ellipsoid is also uniform [78]. A cylinder can be viewed as an ellipsoid with an infinite axis. When the magnetic field in the domain is uniform, for certain shapes of inclusions such as an ellipsoid and a cylinder with circular or square cross-section, it is possible to obtain analytically the equations of eddy current density in the inclusion. EC loss density can be further deduced.
The cylindrical particle is assumed to have infinite length. Thus the problem is reduced to two-dimensional (2D). Two conditions of magnetic field loading are deduced separately.
One case is the exciting field normal to the domain (along cylinder axis); the other is the exciting field in-plane. Finally, a generic formula combines these two cases.
As for the case of spherical inclusion, an exciting field along one axis is first derived. Further, a general formula is generated with arbitrary magnetic field loading. In the end, a general EC loss density formula is given with a shape factor distinguishing the different shapes.

EC Loss Density of SMC

SMC can be viewed microscopically as a periodical layout of inclusions embedded in a dielectric host matrix. Consider now the SMC material as a representative elementary cell containing an inclusion and its surrounding matrix. This cell is supposed to have spatial periodicity. The properties of the inclusion are denoted with the subscript 2 and the properties of the matrix with the subscript 1.
For periodic SMC containing cylindrical or ellipsoidal inclusions, when the filling factor of the inclusion is sufficiently low, each inclusion can be regarded as a single one in the previous section. For biphasic composite, given the effective permeability, μr , the average field in the inclusion is given by (2.15) [88], 〈H〉2 = 1 ξ2(μ2 − μ1) (μr − μ1I) · 〈H〉 (2.15) where 〈H〉 is the average magnetic field over the whole cell. I is the second order identity tensor. And ξ2 is the volume fraction (filling factor) of the inclusions.
The EC loss density U2 in the inclusion can be written as a function of the ‘uniform’ magnetic field in the inclusion (H2 = 〈H〉2): U2 = π2 R2 f σ2μ22H∗ 2 ·K·H2.

Complex Permeability Definition

Complex properties can be used in electromagnetic applications to describe dissipation. A thorough review of homogenization models for dielectric behavior using complex permit2.4 tivity can be found in [42, 66]. 2D and 3D cases have been numerically explored in details.
The complex effective permittivity depends on the properties of each constituent, on their volume fraction and on their spatial arrangement [67]. In an analogous way, complex permeability is a useful tool to handle high frequency magnetic effects, for instance in transformer applications [68, 69]. Power dissipation is directly reflected in the imaginary part of the complex permeability [70, 71]. In the case of SMC at low frequency, when the induced magnetic field can be neglected, there is no time lag between magnetic flux density and magnetic field. In this respect, the imaginary part of complex permeability should be considered as zero. However, EC losses are present – as long as the frequency is not zero.
Thus, an imaginary part, noted μi , can be introduced into the magnetic permeability tensor so as to reflect EC losses.
The complex permeability tensor ˜μ (˜μ = μr − jμi) can be used as a mathematical tool to represent a dissipative magnetic material. In this study, this complex permeability tensor is used to describe the effective properties of SMC. The real part is the usual magnetic permeability, and the imaginary part reflects the EC losses. In what follows, ˜μ denotes the (effective) complex permeability while μr is still called the effective permeability.

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Validation using FEM computations

SMC consists of inclusions surrounded by an insulating film. In this work, attention is focused on two simple microstructures: cubic lattice of spherical inclusions and square lattice of fiber inclusions, as shown in Fig. 2.4. The fiber inclusion problem can be reduced to 2D. Only circular cross-section of fiber is considered The problem of 2D SMC with a magnetic field applied in the normal direction has been discussed in the Appendix A as well as in [86]. At low frequency, the magnetic field is uniform in the domain, so that (2.19) is an exact formula for EC losses. In that simple case, the effective magnetic permeability (real part) is obtained from the Wiener estimate, which also provides in that case an exact value. Therefore, the required validations concern the spherical inclusion case (later referred to as case 1), and the case of cylindrical inclusions with in plane loading (later referred to as case 2). These two cases are associated to more complex field distributions, and require more advanced homogenization techniques.

Table of contents :

List of figures
Résumé en Français
General Introduction
1 Basic Equations for the Modeling of Soft Magnetic Composites 
1.1 Maxwell’s Equations
1.2 Constitutive Relations
1.2.1 Magnetic Behavior
1.2.2 Mechanical Behavior
1.2.3 Magneto-Mechanical Behavior
1.3 Eddy Current Losses
1.3.1 EC Loss Density Definition
1.3.2 EC Loss Density of Homogeneous Structures
1.4 Soft Magnetic Composites
1.4.1 EC Loss Density of SMC
1.5 Homogenization Techniques
1.6 Conclusion
2 A Complex Permeability Model for EC Losses in SMC 
2.1 EC Loss Density of Single Inclusion
2.1.1 Cylindrical Inclusion
2.1.2 Spherical Inclusion
2.2 EC Loss Density of SMC
2.3 Effective Permeability of SMC
2.3.1 MG Estimate
2.3.2 Series Expansion Estimate
2.4 Complex Permeability Definition
2.4.1 Complex Permeability of Single Inclusion
2.4.2 Complex Permeability of SMC
2.5 Validation using FEM computations
2.5.1 Microstructure
2.5.2 Parameters
2.5.3 Effective Permeability Comparison
2.5.4 EC Loss Density Comparison
2.5.5 EC Loss Density by Average Field Assumption
2.5.6 EC Loss Density by Effective Complex Permeability Tensor
2.6 Discussion
2.6.1 Square Microstructure
2.6.2 Cube Microstructure
2.6.3 Complex Permeability
2.7 Conclusion
3 Bounds and Estimates on EC Losses in SMC 
3.1 EC Loss Density in SMC
3.2 Bounds
3.2.1 Cylinder Microstructure
3.2.2 Sphere Microstructure
3.2.3 Extension to More Generic Microstructures
3.3 EC Loss Density Estimates
3.3.1 Cylinder Microstructure
3.3.2 Sphere Microstructure
3.4 Discussion
3.4.1 Numerical Calculations on EC Losses Estimates
3.4.2 Model Validation
3.5 Conclusion
4 Complex Permeability for SMC: Application to Magnetic Circuit 
4.1 EC Loss Density of High Concentration SMC
4.1.1 Perpendicular field
4.1.2 In-plane field
4.1.3 Complex permeability for SMC
4.2 Magnetic Circuit Application
4.2.1 Magnetic Behavior
4.2.2 EC Loss Density
4.3 Conclusion
5 Effect of Stress on Eddy Current Losses in Soft Magnetic Composites 
5.1 Basic Constitutive Equations
5.1.1 EC Loss Density in a Cube-shaped Inclusion
5.1.2 Stress-dependent Magnetic Permeability
5.1.3 EC Losses In a Cube Subjected To Stress
5.2 Loss Density in SMC
5.2.1 Homogenization Technique
5.3 Model Prediction and Results
5.3.1 Material Parameters
5.3.2 Stress Effect
5.4 Conclusion
Conclusion and Perspectives
Appendix A EC Loss Density For Basic Shapes 
A.1 Homogeneous Plate
A.2 Homogeneous Circle
A.3 Homogeneous Square
A.4 Validity Range of Frequency
Appendix B Energy Density 
Appendix C Spherical Symmetry 
Appendix D Mechanical Localization Tensor 


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