FEA, Fundamentals

Eddy Currents Induced by a Time-Varying Magnetic Field

In electric machines, there are two common causes of eddy currents: (1) time-varying currents in coils, and (2) motion of conductors relative to sources of magnetic field. In this post, we show how to estimate the current density arising from a time-varying magnetic field passing through a plate.

Still of current density due to oscillating magnetic field. Results generated using Ansoft Maxwell from ANSYS.

Consider as shown in the figure above a c-shaped iron core with a coil wrapped around one side. The iron core has a small air gap, into which a copper plate is inserted. If we drive a constant current through the coil, we induce a magnetic flux in the iron. The DC flux density across the air gap in our example is about 0.15 T. For a static (DC) field, the copper behaves just like air, not altering the magnetic field.

animation of Eddy currents
Animation of Eddy currents due to harmonic current in coil. Results generated using Ansoft Maxwell from ANSYS.

But when the current and therefore the magnetic field vary harmonically with time, an electromotive force \epsilon is induced according to Faraday’s Law

(1)   \begin{equation*} \epsilon = \oint \vec{E} \cdot \vec{ds} = - \frac{d\Phi}{dt} \end{equation*}

where \vec{E} is the induced electric field, \vec{ds} is an element of the boundary of the area A through which the magnetic flux \Phi passes. We can obtain \Phi by integrating the flux density \vec{B} over the area A according to

(2)   \begin{equation*} \Phi = \iint_A \vec{B} \cdot \vec{n} \, dA \end{equation*}

For our example, the flux density \vec{B} is nearly uniform over the area A and drops off rapidly outside of A, and hence \Phi is well approximated by BA. If B varies harmonically with frequency \omega, then the magnitude of d\Phi/dt and hence the magnitude of the EMF around any loop surrounding the area A is \omega B A.

The current density induced by the EMF \epsilon on a circular loop of radius r in a conductor is given by \epsilon / 2\pi r \rho, where \rho is the resistivity of the conductor. So the magnitude J of the current density is approximately

(3)   \begin{equation*} J = \frac{\omega B A}{2 \pi r \rho$} \end{equation*}

Thus, for our example with B=0.15 T and A=(5 mm)^2 and \rho=1.7e-8 \Omega-m, we expect the flux density just outside the 5 mm square (at r=3 mm) to be 2.8e6 A/m^2 at 40 Hz. This agrees pretty well with the numerical results shown in the animation above, which was generated using Ansoft Maxwell from ANSYS.

This simple approximation works quite well at low frequencies. At higher frequencies, the EMF of the induced currents prevents the magnetic field from penetrating the conductor. The currents drop off rapidly with a skin depth \delta characterized by

(4)   \begin{equation*} \delta = \sqrt \frac{2 \rho}{\omega \mu} \end{equation*}

where \mu is the magnetic permeability of the material. For copper at 40~Hz, we have a skin depth of about 10 mm, much larger then the 1 mm thickness of the plate in our example. We can therefore safely consider the induced current to be uniform through the thickness of the plate.