Eddy currents are generated in a metal workpiece when it is placed within a time-varying magnetic field. Joule heating arises when the energy dissipated by the eddy currents flowing through the workpiece is converted into thermal energy. This heating mechanism is usually referred to as induction heating; the induction cooker is an example of a device that uses this mechanism. The time-varying magnetic field is usually generated by a coil that is placed close to the workpiece. The coil carries either a known amount of total current or an unknown amount of current under a known potential (voltage) difference, alternating at a known frequency.

The eddy current analysis procedure is based on the assumption that a time-harmonic excitation with a certain frequency results in a time-harmonic electromagnetic response with the same frequency everywhere in the domain. In other words, both the electric and the magnetic fields oscillate at the same frequency as that of the alternating current in the coil.

The eddy current analysis provides output, such as Joule heat dissipation or magnetic body force intensity, that can be transferred to drive a subsequent heat transfer, coupled temperature-displacement, or stress/displacement analysis. This allows for modeling the interactions of the electromagnetic fields with thermal and/or mechanical fields in a sequentially coupled manner. See “Mapping thermal and magnetic loads,” Section 3.2.21, and “Sequentially coupled multiphysics analyses using predefined loads,” Section 16.2.2, for details.

Electromagnetic elements must be used to model the response of all the regions in an eddy current analysis including the coil, the workpiece, and the space in between and surrounding them. To obtain accurate solutions, the outer boundary of the space (surrounding the coil and the workpiece) being modeled must be at least a few characteristic length scales away from the device on all sides.

The electromagnetic elements use an element edge-based interpolation of the fields instead of the standard node-based interpolation. The user-defined nodes only define the geometry of the elements; and the degrees of freedom of the element are not associated with these nodes, which has implications for applying boundary conditions (see “Boundary conditions” below).

The electric and magnetic fields are governed by Maxwell's equations describing electromagnetic phenomena. The formulation is based on the low-frequency assumption, which neglects the displacement current correction term in Ampere's law. This assumption is appropriate when the wavelength of the electromagnetic waves corresponding to the excitation frequency is large compared to typical length scales over which the response is computed.

It is convenient to introduce a magnetic vector potential, , such that the magnetic flux density vector . The solution procedure seeks a time-harmonic electromagnetic response, , with frequency radians/sec when the system is subjected to a time-harmonic excitation of the same frequency; for example, through an impressed oscillating volume current density, . In the preceding expressions the vectors and represent the amplitudes of the magnetic vector potential and applied volume current density vector, respectively, while the exponential factors (with ) represent the corresponding phases. Under these assumptions, Maxwell's equations reduce to

The variational form of the above equation is

Abaqus/Standard solves the variational form of Maxwell's equations for the in-phase (real) and out-of-phase (imaginary) components of the magnetic vector potential. The other field quantities are derived from the magnetic vector potential.

The eddy current analysis procedure provides the time-harmonic solution directly at a given excitation frequency. You can specify one or more excitation frequencies, one or more frequency ranges, or a combination of excitation frequencies and ranges.

*ELECTROMAGNETIC, LOW FREQUENCY, TIME HARMONIC
lower_freq1, upper_freq1, num_pts1
lower_freq2, upper_freq2, num_pts2
...
single_freq1
single_freq2
...

*ELECTROMAGNETIC, LOW FREQUENCY, TIME HARMONIC
single_freq1

In a time-harmonic eddy current analysis it is very common that large portions of the model consist of electrically nonconductive regions, such as air and/or a vacuum. In such cases it is well known that the associated stiffness matrix can be very ill-conditioned; i.e., it can have many singularities (Bíró, 1999). Abaqus uses a special iterative solution technique to prevent the ill-conditioned matrix from negatively impacting the computed electric and magnetic fields. The default implementation works well for many problems. However, there can be situations in which the default numerical scheme fails to converge or results in a noisy solution. In such cases adding a “small” amount of artificial electrical conductivity to the nonconductive domain may help regularize the problem and allow Abaqus to converge to the correct solution. The artificial electrical conductivity should be chosen such that the electromagnetic waves propagating through these regions undergo little modification and, in particular, do not experience the sharp exponential decay that is typical when such fields impinge upon a real conductor. It is recommended that you set the artificial conductivity to be about five to eight orders of magnitude less than that of any of the conductors in the model.