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## EC Loss Density of SMC

Eddy currents occur in the ferromagnetic inclusions when the applied magnetic field changes with time. Since the matrix is dielectric, EC occurs only in the inclusions, the macroscopic EC (flowing in the matrix) being negligible. Denote Uinc the EC loss density of the inclusions. From the perspective of the whole composite materials, the EC loss density USMC is then USMC = ξincUinc (1.24) with ξinc the volume fraction of the inclusions.

### 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.

#### Cylindrical Inclusion

Consider that a cylindrical inclusion is placed in infinite space. Set up Cartesian coordinates O x yz by putting the cylinder axis along the z direction and centering the cross-section of the cylinder at O. The cross-section Ω is a disk of radius R, as shown in Fig. 2.1.

Because the cylinder length is infinite, the problem has the property of z-invariance, for instance, ∂z E = 0 and ∂z H = 0.

The magnetic field in the inclusion, H = [Hx , Hy , Hz ]t , can be firstly decomposed into two components: in-plane one, [Hx , Hy , 0]t , and perpendicular one, [0, 0, Hz ]t , where the superscript t is the transpose operator.

**Complex Permeability of Single Inclusion**

Consider that the magnetic field is uniform in the single inclusion: H = Hinc ejωt . The EC loss density can be obtained by the shape-related equation (2.14) or by complex permeability (2.29). Equaling the Setting U = S gives, π Hinc∗ · µinci · Hinc = π2 R2 f σ µ2 Hinc∗ · · Hinc (2.30) =⇒ µinci = πR2 f σ µ2 Therefore, the complex permeability has the form, µ˜inc = µ − jπ R2 f σ µ2 (2.31) which depends on the material properties, working frequency, shape and size (radius).

**Complex Permeability of SMC**

Applying successively the magnetic field in different directions, each element of tensor µi can be obtained. Equaling (2.29) to (2.19), tensor µi can be deduced as, πR2 f σ 2 µ2 µi = 2 µr − µ ∗ µr − µ 1 ) (2.32) ξ2(µ2 µ1)2 − ( 1 ) · ·

Tensor µi is proportional to the frequency f , to the inclusion conductivity σ2 and depends also on the magnetic permeabilities µ1 and µ2 of the constituents, on the volume fraction ξ2, and the shape factor tensor of the inclusions.

Finally, the complex permeability tensor µ˜ for a SMC material is defined as πR2 f σ 2 µ2 µ˜ = µr − j 2 µr − µ ∗ µr − µ 1 ) (2.33) ξ2(µ2 µ1)2 − ( 1 ) · ·

The real part represents the magnetic behavior of the composite, and the imaginary part offers an immediate approach to the EC loss density, by (2.29). It must be noticed that an accurate estimate of the effective magnetic permeability µr is required to define the effective complex permeability tensor µ˜.

In order to determine the EC losses with the complex permeability (imaginary part), a precise estimate of the effective permeability (real part) is essential. The analytical approaches for the effective property of composites with elliptic or ellipsoidal inclusions prove to be arduous. But simple inclusion shapes such as circular cylinder or sphere have1 been widely studied [17, 21, 22, 25]. Thus, in the following, focus is limited on SMC with circular cylindrical or spherical inclusions to verify the application of the complex permeability model with the average field assumption.

**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

**References **