Meshed coil regions

Introduction

Meshed coils in Flux are a type of face (in 2D) or volume (in 3D) region used to represent windings and other current-carrying conductors. As already mentioned in the topic Coils and windings in Flux, these regions provide an interface to an external electric circuit and are fully integrated into the finite element domain. Consequently, Flux can rely on the mesh available in these regions to evaluate relevant electromagnetic quantities (e.g., the magnetic induction, the current density, and the Joule loss density, among others) in the bulk of the conductor.

Since electric currents are sources of magnetic induction, an accurate evaluation of the current density distribution in a coil is of paramount importance in the design of electromagnetic devices. Moreover, the precise computation of this quantity plays a crucial role in the prediction of Joule losses. On the other hand, determining this distribution can be challenging because of well-known current concentration phenomena taking place in conducting media such as the skin and proximity effects.

The Skin and Proximity Effects

When a coil is fed with an electric current, the current density distribution flowing in the bulk of the conductor results non-uniform in the general case.

Several parameters influence the current density distribution in conductors, including:

• the conductivity $\sigma$ of the conductor;
• its magnetic permeability $\mu$ ;
• the time rate of change of the current.

Physicists and Electrical Engineers use the term skin effect to explain current concentration phenomena linked to the parameters above. This term comes from the fact that, in the significant case of conductors subjected to a sinusoidal excitation at a frequency $f$ , a length $\delta$ given by

$\delta = \frac{1}{\sqrt{\sigma \mu \pi f}}$

describes the depth below the conductor surface, concentrating a fixed portion of the current. This length is called the skin depth, and one may show that about 95% (or 1-e^-3) of the total current flows within a depth of three times $\delta$ in a conductor.

The skin effect may be interpreted in terms of an induced current density component flowing in the coil. This induced component modifies the overall current density distribution and opposes to magnetic flux variations dictated by the imposed current, as predicted by Faraday’s law of induction. The example in Figure 1 depicts the skin effect at increasing frequencies in a 1 mm² round copper conductor traversed by a sinusoidal current with an amplitude of 1 mA.

Figure 1. Skin effect affecting the current density distribution in the cross section of circular copper conductor at increasing frequencies or decreasing skin depths

Another phenomenon leading to an uneven current density in conductors is known as the proximity effect. This effect is observed in current-carrying conductors positioned close to each other, as in a tightly wound, multi-turn coil. The proximity effect is also frequency-dependent: it results from an eddy current component induced by the time-varying magnetic flux density field generated by a neighboring conductor. In the case of two conductors traversed by currents with the same orientation, the net effect in each conductor is given by an increased current density in zones away from the other neighboring conductors, as shown in Figure 2. On the other hand, if the currents carried by the conductors have opposing senses, the current concentration will take place in zones closer to the neighboring conductor.

Figure 2. Proximity effect affecting the current density distribution in the cross sections of two neighboring, parallel round conductors. In this example, both conductors are traversed by a sinusoidal current flowing in the same sense.

Homogenization Techniques in Flux

Evaluating losses in thin multi-stranded conductors or coils with several turns with the finite element method can be a demanding task, since the method requires meshing devices with detailed geometries. A well-known rule of thumb recommends that the zone corresponding to the first two or three skin depths below the conductor surface should be finely discretized. However, satisfying this criterion frequently leads to complex meshes and high memory requirements, especially in 3D and at higher frequencies.

Flux deals with this trade-off between mesh complexity, use of computer resources, and accuracy by providing the user with two types of Meshed Coil regions: the solid conductor region and the coil conductor region.

Solid conductor regions are employed in Flux to represent an individual current-carrying conductor or a coil in its full geometrical complexity. They can take into account the skin and proximity effects rigorously and are useful when a detailed mapping of the current density distribution in the conductor is desired. On the other hand, they require elaborate geometrical and circuit descriptions and lead to complex meshes, a substantial memory usage, and relatively longer computation times.

Coil conductor regions, on the other hand, are well suited to model tightly packed windings, in which the positioning of the wire turns is given by the juxtaposition of a fixed motif or unit cell. In these regions, Flux performs a pre-computation on a unit cell to estimate the impact of the skin and proximity effects in the current density distribution. The results yielded by this pre-computation are then translated into equivalent material properties that are assigned to the coil conductor region, to have more informations about these elementary cells, see this discussion.

These regions can also descbribe foil coil windings (wound metallic conducting sheet). In this homogenization type Flux compute with a dedicated fomulation the impact of the skin and proximity effects in the current density distribution. Thereby, as there are not the same type of modelisation, the available quantities in the postprocessing could be differents, to have more informations about this model, see this discussion.

The techniques described above homogenizethe winding representation (i.e., simplifies its geometrical description) and circumvents the previously mentioned difficulties related to solid conductor regions. However, it should be remarked that the equivalence between the non-homogenized and homogenized representations is not absolute. While the homogenization preserves the Joule losses and the stored magnetic energy, the evaluated current density distribution changes. In other words, the current density computed in a homogenized coil conductor region is not identical to the current density distribution predicted by a coil model based upon solid conductor regions.

The example in the next section compares the modeling of a simple winding with both solid conductor regions and a coil conductor region to further clarify these aspects.

Comparing Solid Conductor Regions and Coil Conductor Regions in Flux

To compare the use of Solid conductor and Coil conductor regions, let us consider the modeling in Flux 2D of a hypothetical induction heating device consisting of a 20-turn coil magnetizing a linear ferromagnetic structure and a copper plate. The coil itself is formed from a copper conductor with a circular cross-section and fed by a current source at increasing frequencies.

Figure 3 displays a first modeling approach for this device based upon the use of several solid conductor regions. The twenty turns of the coil are explicitly represented in the geometry, and each circular surface associated with a turn is assigned to an individual solid conductor region. Moreover, each solid conductor region is linked to its own FE coupling component (2-terminal solid conductor type). In the external circuit, the twenty FE coupling components must be connected in series to the source, since all turns are expected to carry the same current.

Figure 3. A Flux 2D project modeling a 20-turn coil in a hypothetical induction heating device operating at 1 kHz. Each coil turn is assigned to a Solid Conductor region, and their current density distributions affected by the combined skin and proximity effects are shown in (a). The coupled electric circuit containing one FE coupling component for each Solid Conductor region is shown in (b).

Figure 4, on the other hand, displays the same device modeled with a different approach. The twenty turns are no longer represented individually in the geometry and a single coil conductor region is assigned to a rectangular surface enclosing the winding. Consequently, only one FE coupling component (Stranded coil conductor type) is required, and both the geometry description and the representation of the external circuit become simpler.

Figure 4. The same induction heating device operating at 1 kHz, but now modeled in Flux 2D with a single Coil Conductor region. In this approach, Flux evaluates a uniform equivalent current density in the Coil Conductor region. On the other hand, the geometrical description of the model and the external circuit (b) become straightforward.

Summary

The table below summarizes the most important concepts analyzed in the previous sections. It may be conveniently used as a quick reference guide for the modeling of coils and other lossy conductors in Flux.

Table 1. Comparison of coil modeling approaches in Flux.

Meshed Coil region type	Current density	Mesh	Description of the coupled electric circuit	Frequency behavior of Joule losses
Solid Conductor	Flux evaluates a non-uniform current density distribution reflecting the skin and proximity effects.	Complex meshes required.	Potentially elaborate. One FE coupling component required for each Solid Conductor region.	Skin and proximity effects fully taken into account, leading to augmented Joule losses at higher frequencies.
Coil Conductor with a detailed geometrical description	Flux evaluates an equivalent uniform current density distribution.	Coarser meshes are possible.	Straightforward. Only one FE coupling component required.	Skin and proximity effects taken into account by means of homogenization of the winding. Augmented Joule losses at higher frequencies as in the case of Solid Conductor regions.
Coil Conductor with a simplified geomerical description				Joule losses are non-zero and constant at any frequency. Results are accurate for direct current and low-frequency excitations.
Coil Conductor Without Losses				Joule losses are constant and equal to zero at any frequency.

How to Use Meshed Coil Regions in Flux

Bibliographical references

For further reading on the skin and proximity effects, on the evaluation of losses in coils and on homogenization techniques, a list of bibliographical references is available here.