# Coil conductor region representing a foil coil

## Introduction

This chapter discusses the creation of a coil conductor region with losses and detailed geometrical description to represent a foil coil.

The following topics are covered in this documentation:
• What this type of region models.
• How to create a coil conductor region describing a foil coil in a Flux project.
• Limitations.
• Example of application.

## What this type of region models

A foil coil is a winding obtained from a thin, rectangular, metallic sheet folded in a spiral-like shape, as shown in Figure 1. The sheet is covered by an insulating coating (varnish). This kind of coil design is common in electromagnetic devices such as power transformers and reactors.

The current density distribution in a foil-wound coil fed by a time-varying source depends on skin and proximity effects. Since the foil is usually very thin and made from a material with a high electrical conductivity, the skin effect along its thickness is negligible (i.e., the current density in each turn results practically uniform along a radial direction). On the other hand, the current density in each foil turn may greatly vary along the axial direction of the coil as a function of both position and frequency.

This anisotropic behavior is specific to foil coils and influences the Joule losses developed in the bulk of the coil material. Thus, Flux implements a special homogenization technique to efficiently represent it in its 2D Steady State AC application. This technique is exclusive to foil coils and differs from the approach used in the other subtypes of coil conductor regions with losses and detailed geometric description representing stranded coils. Further details on this homogenization technique may be obtained through the bibliographical references section.

## How to create it in a Flux project

The foil-wound coil is a subtype of coil conductor region with losses and detailed geometrical description that is only available in Flux 2D. The availability of these regions in Flux FEM applications is discussed in the following documentation topic: Coil models and their availability in Flux projects.

This region may be created as follows:
• While creating a new region, select Coil Conductor Region in the drop-down menu Type of region.
• In the Basic Definition tab, proceed in the same manner as in the case of a coil conductor region without losses.
• In the Coil Loss Models tab, proceed now in the same manner as in the case of a coil conductor region with losses and simplified description, but select Detailed description (considers proximity and skin effects) in the drop-down menu instead of Simplified description (neglects proximity and skin effects). This action will display the Strand or Unit Cell definition drop-down menu.
• In the Strand or Unit Cell definition drop-down menu, select the type: Foil-wound coil. The template is displayed in Table 1.
• Provide the geometrical parameters required for characterizing the foil coil, in accordance with Table 1.
Table 1. Template for coil conductor region with losses and detailed geometrical description modeling a foil coil winding.
Type 2D foil coil representation Required parameters
Foil-wound coil • d: sheet thickness.

## Limitations

For a given frequency f, the homogenization approach assumes that the foil thickness d is always smaller than the skin depth δ. In practice, this hypothesis restrains the validity of the model with an upper frequency limit. Thus, the model yields good results in a frequency range not exceeding a maximum frequency fmax given by

${f}_{\text{max}}=\frac{\rho }{\pi µ}{\left(\frac{N}{\lambda t}\right)}^{2}$

In the previous expression:
• $µ$ is the magnetic permeability of the foil material,
• $\rho$ is the electrical resistivity of the foil material,
• $N$ is the number of turns of the coil,
• $\lambda$ is the coil fill factor and
• $t$ is the thickness of the winding.

As for Coil Conductor region with losses and simplified geometrical description, the user may post-process quantities related to the material resistivity in the surface (2D) representing the coil (e.g., the power loss density in the winding or the total dissipated power).

Consequently, the Joule losses dissipated by the coil may be evaluated in the same manner as: Coil Conductor region with losses and simplified geometrical description.

Important: The following remarks and limitations apply:
• The foil orientation u (or the direction of the coil axis) must be parallel to the Oy direction of the global coordinate system, as shown in Table 1.
• Coil conductor regions representing foil coils may only be assigned to rectangular surfaces.
• This coil model does not support the additional resistance described in the circuit component: Coil conductor component associated to the region.
• The foil turns are considered in series. Their total number is provided the Basic definition tab.

## Example of application

The foil coil configuration shown in Figure 2 has been analyzed in the article Calculation of Current Distribution and Optimum Dimensions of Foil-Wound Air-Cored Reactors by M.M. El-Missiry (Proceedings of the Institution of Electrical Engineers, vol. 124, no. 11,November 1977, DOI: 10.1049/piee.1977.0218 ). In that work, the author presents a circuit-based, semi-analytical method to compute the current density distribution and several other electromagnetic quantities of a foil coil.

The coil in Figure 2 may be easily modeled in Flux 2D with the foil coil template available for coil conductor regions with losses and detailed geometrical description, as described in the previous sections of this chapter. Figure 3 shows the results obtained with an axisymmetric Steady State AC Magnetic application at 50 Hz and with an additional horizontal symmetry (i.e., only one quarter of the foil coil is represented). The development of a non-uniform current distribution pattern characteristic to foil coils may be verified in the color plot available in that figure.

A comparison between the current density results obtained with the approach described in that article and the solution evaluated with Flux 2D is provided in Figure 4. The graph in this figure displays the real and imaginary parts of the current density phasor (in RMS values) on a path from its upper extremity (0.0 p.u.) to its center (0.5 p.u.) along one of the centermost turns of the coil (as depicted in Figure 3

An additional comparison between measured lumped circuit parameters (provided in El-Missiry's article) and their corresponding values computed with Flux 2D (obtainable, for instance, with the help of I/O Parameters defined by formulas) is available in Table 2.
Table 2. Comparison between resistance and reactance measurements and the results yielded by Flux 2D for the Aluminum foil coil represented in Figure 2.
Lumped circuit parameter at 50 Hz Measurement Flux 2D Deviation
Reactance 1.802 Ω 1.827 Ω 1.39%
Resistance 0.382 Ω 0.376 Ω 1.57%
The results from Figure 4 and from Table 2 show that the FEM solution evaluated with Flux 2D is in excellent agreement with both measurements and other numerical techniques.

## Bibliographical references

For further information on the foil coil homogenisation approach implemented in this feature, please refer to the bibliographical references chapter.