# Solid conductor regions representing coils

## Introduction

This chapter discusses the creation of solid conductor regions in a Flux project. This type of entity is employed in Flux to accurately represent the current flow (imposed and induced) in a conductive medium that may or may not be connected to an electric circuit. The specific case corresponding to 2-terminal solid conductor regions is particularly relevant to the modeling of coils taking part in electromagnetic devices and will be focused in this chapter.

The following topics are covered in this documentation:
• What this type of region models.
• How to create a 2-terminal solid conductor region representing a coil in a Flux project.
• Limitations.
• Example of application.

## What this type of region models

Solid conductor regions are conceived to represent the general case of a conducting medium that may be fed by an external circuit through a single input and a single output (in 2D and 3D) or through a larger number of connections (in 3D). Thus, this kind of region may be conveniently used to represent coils in a Flux project.

From a finite element method point of view, solid conductor regions are characterized by their ability to account for two superposed current distributions: one imposed and the other induced. In this type of region, these components are determined directly from the underlying finite element formulation during resolution.

The total current density distribution corresponding to these combined components results non-uniform across the region in the general case. Consequently, solid conductor regions are able to accurately represent the skin and proximity effects in either Steady State AC or Transient Magnetic applications.

This is in contrast with the behavior of coil conductor regions with losses and detailed geometrical description, which are dedicated to the representation of windings constructed from thin wires and rely on homogenization to account for the skin and proximity effects. Even if the evaluated current density distribution in this type of region remains uniform, the homogenization approach allows an accurate evaluation of the increased losses due to skin and proximity effects.

Thus, solid conductor regions are well adapted to user applications requiring a refined determination of local quantities such as the distributions of current density, magnetic flux density, loss density and temperature in conductive media of arbitrary shape. Moreover, they may also be used to represent a coil on a turn-per-turn basis, especially in the case of wires with unusual cross-section shapes or in windings with specific packings that do not match any of the unit cell templates available for coil conductor region with losses and detailed geometrical description.

In any of these cases, a trade-off exists between the accuracy provided by the solid conductor region and the increased complexities of geometry, mesh and coupled circuit resulting from their use. An in-depth, comparative discussion on this subject is available in this documentation topic: Comparing Solid Conductor Regions and Coil Conductor Regions in Flux.

## How to create it in a Flux project

In Flux 2D and in Flux Skew, the solid conductor region is a face region, while in Flux 3D it becomes a volume region. The availability of these regions in Flux applications is discussed in the following documentation topic: Coil models and their availability in Flux projects.

In Flux 2D and in Flux Skew, this region may be created as follows:

• while creating a new region, select Solid Conductor Region in the drop-down menu Type of region;
• Then, provide a material for the solid conductor;
Note: The provided material must have both an electrical property and a magnetic property defined. In other words, it must contain the models for the constitutive relations J(E) and B(H) that relate the current density J to the electric field E and the magnetic flux density B to the magnetic intensity H, respectively. The simplest approach would be adopting a material with constant, isotropic resistivity ρ = 1/σ and magnetic permeability µ in such a way that J = σE and B =µH. Other constitutive relation models are available in Flux, and the user could also rely on Flux Material Manager to import predefined materials in the Flux project.
• Inform how the solid conductor is connected to an external source in the Circuit Coupling drop-down menu. Three possibilities are available:
• Circuit defined: the solid conductor region is linked to a FE coupling component of type 2-terminal solid conductor integrating an external circuit. If this option is chosen, the corresponding FE coupling component in the circuit and the orientation of the current (positive or negative) must also be provided.
• No circuit: open circuit conductor: the solid conductor region is considered as disconnected from any external circuit.
• No circuit: conductor in parallel with all others of the same type: the solid conductor region is also considered disconnected from any external circuit. However, its input terminal is considered to be short-circuited to the input terminals of other solid conductor regions (i.e., all input terminals are set to the same potential) in the finite element domain. Similarly, its output terminal is considered to be short-circuited as well to the output terminals of the other solid conductor regions.
Note: Even if the solid conductor region is set to one of the No-circuit connection modes above, an induced current component may still flow. The specific type of No circuit connection will affect the path of the eddy current loops induced in the bulk of the solid conductor. Further information is available in the following documentation topic: No-circuit solid conductors description in 2D.
Note: In the case of a solid conductor region in a symmetrical or periodical domain, the symmetry and periodicity options must be configured during the creation of the FE coupling component. This workflow is slightly different from the case of the other meshed coil regions (i.e., coil conductor regions), for which the symmetry and periodicity behaviors are configured during the creation of the region.
In Flux 3D, the procedure to create a solid conductor region is similar to the one described above for Flux 2D and Flux Skew. The few remarkable differences are related to an eventual connection of the region to an external circuit through its two terminals:
• To complete the creation of a solid conductor region in Flux 3D, the user must provide only the material of the conducting medium.
• After completing the creation of the solid conductor region, the user must successfully assign it to the corresponding volumes in the finite element domain.
• The link between the created solid conductor region and the corresponding FE coupling component in an existing electrical circuit is performed in a last step. To complete it:
• Go to the menu Physics and select Assign terminals to solid conductors → Assign terminals to solid conductors 2 terminals (completion mode) or Assign terminals to solid conductors → Assign terminals to solid conductors 2 terminals (modification mode).
• At this point, Flux will ask for the faces in the geometry corresponding to the input and output terminals of the FE coupling component accordingly to the selected method. Further information on the assignment of terminals to a solid conductor is available in the following documentation topic: Describing active solid conductors (3D).
• The step above is only mandatory for a solid conductor region connected to an electrical circuit. If that step is not performed, the solid conductor region will be considered disconnected from the circuit and will be subjected only to induced currents.

## Limitations

Solid conductor regions are not available in Magnetostatic applications (neither in Flux 2D, nor in Flux Skew nor in Flux 3D).

However, in Flux 3D, the user may emulate the behavior of a conductor subjected to DC current with a solid conductor region in a Transient or in a Steady State AC application by disabling the evaluation of the induced current component predicted by the finite element formulation. This is accomplished by checking the option "Eddy currents not considered during the solving process (only DC current)" during the creation of the solid conductor region. Consequently, with this option both skin and proximity effects will not be considered in the resolution.

In Flux 3D, the shape of a volume associated to a solid conductor region is constrained to simply connected topologies. Non-simply connected volumes may be assigned to solid conductor regions with the help of Electric loop cuts. For further information on this subject, please refer to the following documentation chapter: Cut of electric loop.

## Example of application

Figure 1 shows a 4-turn induction heating coil formed from a hollow rectangular copper conductor. A power amplifier feeds the coil with high-intensity, high-frequency sinusoidal current. Consequently, the time-varying magnetic flux density established by the coil induces an eddy current in the bulk of an Aluminum pipe, which is also conductive and thus heats by Joule effect.

Such an inductive heating system was modeled in Flux 2D in a Steady State AC Magnetic Axisymmetric application. Both the induction coil and the Aluminum pipe were represented by solid conductor regions. While the solid conductor regions corresponding to the coil are fed by an electric circuit representing the power amplifier, the solid conductor region associated with the pipe is disconnected and is only subjected to the induced currents arising from the finite element computation.

Figure 2 shows the current density distributions in the coil and in the pipe for an input current of 700 A rms at 500 Hz (a relatively low frequency for induction heating applications).

In Figure 2, the skin and proximity effects are accurately accounted for in the solid conductor regions, both in the pipe and in the coil. However, at higher frequencies of operation, the current concentration becomes even more pronounced and meshing the solid conductor regions to an element size compatible with the skin depth might become challenging.

Under such circumstances, it may be advisable to adopt the special solid conductor region implementing a surface impedance boundary condition, which is available in Flux 3D. Figure 3 shows a Flux 3D project based on this latter approach and representing a similar induction heating device operating at a much higher frequency of 100 kHz in a coupled Steady State AC Magnetic - Transient Thermal application.

For further information on the special solid conductor region implementing the surface impedance boundary conditions, please refer to the following documentation topic: Surface impedance condition (3D).