Eddy Currents 3d

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Eddy Currents in 3DSOLVED WITH COMSOL MULTIPHYSICS 3.5a

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Eddy Cu r r e n t s i n 3D

Introduction

Induced eddy currents and associated thermal loads are of interest in many high power AC applications. This example is of general nature and illustrates some of the involved physics as well as suitable modeling techniques in the AC/DC Module.

Model Definition

A metallic plate is placed near a 50 Hz AC conductor. The resulting eddy current distribution in the plate depends on the conductivity and permeability of the plate. The model considers four different materials: copper, aluminum, stainless steel, and magnetic iron.The geometry consists of a single wire and a plate with dimensions as shown below.

10 mm

60 mm

100 mm

24 kA, 50 Hz

100 mm

E D D Y C U R R E N T S I N 3 D | 1


Because you cannot afford meshing an infinite volume, it is necessary to specify a finite volume to mesh and solve for. In this case, it makes sense to enclose the wire and the plate in a cylinder with the wire on the axis of this cylinder.

The conductor is modeled as a line current with 0° phase and an effective value of 24 kA.

In the subdomains the magnetic vector potential is calculated from

where σ is the conductivity, ε the permittivity, and μ the permeability.

An important parameter in eddy current modeling is the skin depth, δ.

jωσ ω2ε–( )A ∇ 1μ---∇ A×⎝ ⎠⎛ ⎞×+ 0=

δ 2ωμσ-----------=



The following table lists the skin depth for the different materials at a frequency of 50 Hz.

The combination of a high permeability and a high conductivity can make it impossible to resolve the skin depth when modeling. When the skin depth is small in comparison to the size of conducting objects, you can exclude the interior of those objects from the model and replace them with an impedance boundary condition accounting for induced surface currents. This condition uses the following relation between the magnetic and electric field at the boundary:

The distribution of the dissipated power, Pd, has the unit W/m2, and can be calculated from

where JS is the induced surface current density, and the asterisk denotes the complex conjugate.

Alternate Modeling Technique Using Reduced Form

It is possible to reduce the size of the geometry in this model to only include the plate because the field from the straight wire can be represented by an analytical solution. The model needs to predict the perturbations in that field caused by the presence of the plate. An important modeling assumption is that the presence of the plate in the vicinity of the wire does not affect the current in the wire.

The magnetic vector potential around an infinitely long wire surrounded by air, oriented along the x-axis and carrying a current J is given by

MATERIAL REL. PERMEABILITY CONDUCTIVITY SKIN DEPTH

Copper 1 5.998·107 S/m 9 mm

Aluminum 1 3.774·107 S/m 12 mm

Stainless steel 1 1.137·106 S/m 67 mm

Iron 4000 1.12·107 S/m 0.34 mm

n H× ε jσ ω⁄–μ

----------------------n E n×( )×+ 0=

Pd12--- JS E*⋅( )=

Axμ0J rlog

2π----------------------=



where r is the distance of the field point from the wire.

The following plot shows the reduced modeling geometry. Infinite elements are used in the outer spherical shell to further reduce the size of the computational domain.



Results and Discussion

The induced eddy current distribution for a plate made of copper is shown as streamlines, whereas the distribution of the ohmic losses is shown as a slice plot.

A total dissipated power of 6 W was obtained from integration through the plate. By repeating the simulation for different materials, the model shows that lowering the conductivity decreases the dissipated power. However, for high permeability materials like soft iron, the dissipated power is higher than in a copper despite a much lower conductivity.

The corresponding reduced form solution to the model with the iron plate is shown below.



It is possible to compare the results from the full model with those from the reduced form model. The total losses in the iron plate is a good measure to use for comparison, and the results predicted by the two models are fairly close (approximately 27 and 28 W, respectively). The difference can be accounted for by the fact that the reduced model represents the field from the wire to a higher accuracy because this field is given as an exact solution. Also, because the modeling domain is smaller, it is possible to use a finer mesh with no increase in memory usage.


Eddy Currents 3d

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