Magnetostriction

2D model of inductor with magnetostriction.
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Introduction

To run the model, open choke.pro with Gmsh.

This 2D model compute the mechanical deflection of a two collumns inductor. It take into account of:

• The magnetostriction effect in magnetic sheets
• The Maxwell stress tensor into the whole domain (Airgap and magnetic core)

Magnetic and mechanical physics are weakly coupled and assume that mechanical deflections doesn't modify the magnetic field. In this way, Magnetic field is first computed, and the Maxwell stress tensor as well as the Magnetostriction strain tensor are deduced from this simulation. This two tensors are injected in the mechanical simulation and deflection can be obtained

The mecanical model allows also to estimate the different resonant frequencies or the inductor

file details:

• choke.geo: the geometry which is parametrized
• choke.pro: Pro File associated to choke.geo
• Elasticity_2D.pro: The mechanical physics
• magnetostriction.txt: The magnetostriction curve
• jacobian.pro: The methods of integration
• Magsta2D.pro: The electromagnetic physics

Geometry

A two columns inductor is modelized and the geometry can be modified using parameters:

• Column height
• Column width
• Number of airgap
• airgap thichness
• yoke length
• winding thickness

This allow the possibility to model different kind of inducor easily

Boundaries

Magnetical

The magnectic vector potential a is fixed to 0 for the external boundaries of the system. The current density is imposed into the winding and deduced thanks to the nominal current, the number of turns and the cross section of the winding.

Mechanical

The inferior yoke is assumed fixed u=0

Magnetics

A 2D Magnetostatic solver is used for this study. the model compute the The magnectic vector potential a. Materials are considered linear and sources of currents are directly imposed. The solver is developped into MagstA2D.pro

\( \begin{eqnarray} \nabla^2 \mathbf{A} + \mu_0 \mu_r \mathbf{j_z} = \mathbf{0} \label{eq:vector_potential} \end{eqnarray} \)

Different post-proccessing are integrated as:

• The Flux density B
• The Magnetic field H
• The energie per length unit (J/m)

Maxwell Stress Tensor

The Maxwell Stress Tensor is deduced directly from the magnetic field and is injected into the mechanical solver.

\(\begin{eqnarray} \sigma_{mst}= \frac{1}{\mu_0\mu_r}\begin{bmatrix} B_x^2 -\frac{1}{2}B^2 & B_xB_y\\ B_xB_y & B_y^2 -\frac{1}{2}B^2 \end{bmatrix} \label{eq:stress} \end{eqnarray}\)

The following code in used :

 sig_maxwell[]=Vector[ CompX[$1]*CompX[$1]-Norm[$1]*Norm[$1]/2,
		       CompY[$1]*CompY[$1]-Norm[$1]*Norm[$1]/2,
		       CompX[$1]*CompY[$1]];

Magnetostriction strain tensor

The magnetostriction tensor is obtained using the flux density map and with the magnetostriction curve of the material witch link the strain with the flux density. This kind of curve is presented in the next figure

The curve permit to determine the strain tensor in the referentiel (Bt,Bn) presented in the next figure.

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The strain tensor is decomposed into two phenomenas: the normal and tangential magnetostriction, called respectively \(\lambda_N\) and \(\lambda_T\). \(\lambda_T\) could be obtained using experimental data, or by asuming that magnetostriction doesn't modify the volume, a simple relation is obtained\[\lambda_N=-\lambda_N/2\]

\( \begin{eqnarray} \epsilon_{(B_t,B_n)}=\begin{bmatrix} \lambda_T & 0 \\ 0 & \lambda_N \end{bmatrix} \label{eq:strain_tensor1} \end{eqnarray} \)

In order to be injected into the mechanical model, the strain tensor needs to be converted into the (x,y) basis. This is done by computing the change of basis matrix P. In this way the strain tensor in the basis (x,y) is determinated.

\( \begin{eqnarray} P=\begin{bmatrix} \frac{B_x}{B} & -\frac{B_y}{B} \\ \frac{B_y}{B} & \frac{B_x}{B} \\ \end{bmatrix} \label{eq:P} \end{eqnarray} \)

\( \begin{eqnarray} \epsilon_{(x,y)}=P \epsilon_{(B_t,B_n)}\ P^t \label{eq:PPT} \end{eqnarray} \)

\( \begin{eqnarray} \epsilon_{(x,y)}=\begin{bmatrix} \frac{1}{B^2}(\lambda_T B_x^2+\lambda_N B_y^2) & \frac{B_xB_y}{B^2}(\lambda_T-\lambda_N) \\ \frac{B_xB_y}{B^2}(\lambda_T-\lambda_N) & \frac{1}{B^2}(\lambda_T B_y^2+\lambda_N B_x^2) \\ \end{bmatrix} \label{eq:strain_xyz} \end{eqnarray} \)

The following code in used :

lamb[]=Vector[lambdap[Norm[$1]],lambdaper[Norm[$1]],0];

sig_vect[]=C_m[]*lamb[$1];

sig_mat[]=Tensor[CompX[sig_vect[$1]], CompZ[sig_vect[$1]]  ,   0,
	         CompZ[sig_vect[$1]], CompY[sig_vect[$1]]  ,   0,
		            0	    ,          0	   ,   0 ];

// Change of basis

P[]=Tensor[ CompX[$1]/Norm[$1]  ,   -CompY[$1]/Norm[$1] , 0,
            CompY[$1]/Norm[$1]  ,    CompX[$1]/Norm[$1] , 0,
                       0        ,             0         , 1 ];
PP[]=Transpose[P[$1]];
sig_PPP[]=P[$1]*sig_mat[$1]*PP[$1];
sig_magnetostriction[]=Vector[CompXX[sig_PPP[$1]],CompYY[sig_PPP[$1]],CompXY[sig_PPP[$1]]];

Mechanical

A 2D mechanical solver is added to the model. this harmonic solver permit to compute the deflection of the inductor due to the magnetostriction and the maxwell stress tensor. It permits also to obtain the eigen modes of the magnetic core. No Damping is present added so that resonnance are not

Resolution

4 resolution methods are provided:

• A magnetostatic study
• A magneto-mechanical study with allow to estimate the complete deflection of the magnetic core due to magnetic effects
• A magneto-mechanical study on a specific point. It compute the deflection of a specific point. It is the same procedure as the previous point but allow easily to compute the FRF on a specific point. It show that magnetostriction and/or Maxwell dtress tensor can stimulate a resonnance
• A mechanical study to obtain the eigen mode od the inductor.

the magnetomechanical model start firsty to compute the magnetic field. Them it compute the magnetostriction and the maxwell tensor for the mechanical model. At the end the mechanical model is running.

References

Models developed by Mathieu Rossi, Jean Le Besnerais and Christophe Geuzaine. Copyright (c) 2015 EOMYS