Coil Conductor region with losses and detailed geometrical description

Introduction

This chapter discusses the creation of coil conductor regions with losses and a detailed geometrical description. This type of coil region requires an in-depth description of the winding geometry, allowing Flux to evaluate frequency-dependent Joule losses.

The following topics are covered in this documentation:

What this type of region models.
How to create a coil conductor region with losses and detailed geometrical description in a Flux project.
Limitations.
Example of application.

What this type of region models

The coil conductor region with detailed geometrical description allows the user to represent a coil in the finite element domain. The region behaves as a magnetic field source and may be either driven by a coupled circuit or have its current imposed by the user.

This type of region may be regarded as an extension of the coil conductor region with losses and simplified geometrical description, in which the user must provide additional parameters to fully characterize the geometry of a unit cell within the coil. This unit cell is defined as the smallest repeated pattern of the winding, as shown in Figure 1

Figure 1. Definition of a unit cell for windings created from (a) rectangular and (b) circular conductors.

The coil conductor region with losses and detailed geometrical description may be used to represent single or multi stranded conductors with rectangular or circular cross sections. Consequently, the coil conductor region with detailed geometrical description may be used to represent the most frequent types of coils occurring in electromagnetic devices.

Note: They may also be employed to represent foil coils, which are obtained from a wound metallic tape or sheet and are discussed in a different documentation topic. To have more informations about this model, see this discussion.

The full characterization of the unit cell allows Flux to account for frequency-dependent current concentration phenomena such as the skin and proximity effects through homogenization. Hence, the Joule losses evaluated in a coil conductor region with detailed geometrical description are frequency-dependent as well, making this region type well adapted to the modeling of coils and windings operating in wide frequency bands. For a comparison of the frequency behaviors of the Joule losses in each kind of coil conductor region available in Flux, please refer to the following documentation topic: Comparing Solid Conductor Regions and Coil Conductor Regions in Flux.

Moreover, the coil conductor region with detailed geometrical description also supports an additional resistance, which is provided as a lumped resistance value while creating its associated FE coupling component.

For further details on related types of coil conductor regions, the user is referred to the following documentation topics:

How to create it in a Flux project

In Flux 2D and in Flux Skew, the coil conductor region with detailed geometric description is a surface region, while in Flux 3D it becomes a volume region. The availability of these regions in Flux FEM applications is discussed in the following documentation topic: Coil models and their availability in Flux projects.

In any case, 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, the Strand definition tab and the Orientation & units tab.
In the Strand or Unit Cell definition drop-down menu, select a type of unit cell from the list. The available templates are displayed in Figure 1 and in Table 1
In the Strand definition tab, provide the geometrical parameters required for characterizing the unit cell, in accordance with Table 1
In the Orientation & units tab, the units affecting the geometrical parameters of the Strand definition tab may be modified. The coordinate system and vector u orienting the unit cell (as shown in Table 1) may be modified as well, if required. For further information on the procedure for orienting the wires in 3D, please refer to the following documentation page: Orienting a Coil Conductor region in 3D.

Table 1. Unit cell templates available for coil conductor region with losses and detailed geometrical description.
Type	Unit cell representation	Required parameters
Rectangular section wire		dh: horizontal dimension of the strand; dv: vertical dimension of the strand; k = gh/gv: ratio of horizontal to vertical gaps between strands; n: number of strands in parallel.
Circular section wire: diameter		d: strand diameter; n: number of strands in parallel
Circular section wire: fill factor		K_f: fill factor n: number of strands in parallel.

Note: A common parameter of each unit cell template is the number of strands in parallel. This parameter allows the user to describe windings in which a wire with a larger cross section has been replaced by a bundle of n smaller wires connected in parallel (i.e., a multi-strand wire), to decrease losses by skin effect while guaranteeing that the same total current flows in the coil. The total number of unit cells in the finite element region is given by:

$N_{c e l l} = N t n$ With Nt the number of turns (defined in the Basic Definition tab) and n the number of strands in parallel

Changing the number of strands in parallel modifies the winding in accordance with Figure 2. In the coil regions displayed in that figure, the number of turns is 15 in both cases. However, while the number of strands in parallel is equal to one in (a), the number of wires in parallel was changed to seven in (b).

Figure 2. A coil conductor region with losses and detailed geometrical description with 15 turns and a single strand in parallel (a). The same region with 15 turns, but with seven strands in parallel instead (b). Note that for this kind of coil, each strand in parallel is crossed by the same current, if it is not the case, please consider the use ofSolid conductor regions representing coils.

Limitations

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 (in 2D) or volume (in 3D) regions 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 th same manner as:Coil Conductor region with losses and simplified geometrical description

In order to have good results, it is necessary to respect the following hypothesis required by the homogenization technique:

A coil described by this type of region must have several patterns representing the elementary cells (strands) in both directions of the current cross-section. It is necessary to have at least ten elementary cells in each direction as shown in the following example figure: Figure 3

Therefore, this type of region cannot model devices such as planar coils or hairpin coils because they have too few elementary cells in one of the both direction.

This type of region is not available for the 2D axysimmetric magnetic transient application of Flux.

For the transient application, the limit frequency is setted to 100kHz.

Note: In case the device to be modeled does not respect this limitation, the solid conductor region represents a good alternative to this coil conductor region as shown in this section: Comparing Solid Conductor Regions and Coil Conductor Regions in Flux.

Example of application

Let's consider the coil shown in Figure 3

Figure 3. An example of coil geometry befitting the coil conductor region with losses and detailed geometrical description.

In his seminal work A Treatise on Electricity and Magnetism, J.C. Maxwell provided an approximation for the self-inductance of such a coil. Maxwell's inductance formula is

$L = (4 π {. 10}^{- 7}) r_{mean} n^{2} [l n (\frac{8 r_{mean}}{R}) - 2]$ (1)

In the expression above, n is the number of turns and r_mean is the mean radius of the coil. R is a parameter known as the geometrical mean radius of the rectangular cross section:

$l n R = l n (\sqrt{a^{2} + b^{2}}) - \frac{1}{6} \frac{a^{2}}{b^{2}} l n (\sqrt{1 + \frac{b^{2}}{a^{2}}}) - \frac{1}{6} \frac{b^{2}}{a^{2}} l n (\sqrt{1 + \frac{a^{2}}{b^{2}}}) + \frac{2}{3} \frac{a}{b} \tan^{- 1} (\frac{b}{a}) + \frac{2}{3} \frac{b}{a} \tan^{- 1} (\frac{a}{b}) - \frac{25}{12}$ (2)

In Maxwell's approach to derive these expressions, the current is supposed uniform over the cross section of the coil. Thus, equation (1) neglects the impact of the skin and proximity effects in the self-inductance of the coil at higher frequencies. However, this dependency may be modeled in Flux. The coil conductor region with losses and detailed geometrical description is well-adapted for such an investigation.

Indeed, Figure 3 contains a complete geometrical description of the coil, including the data required for a full characterization of the unit cell of the winding. In this example, the user could choose either the Circular section wire: diameter or Circular section wire: fill factor templates described in Table 1 for creating the coil region. The inductance may be evaluated with a sensor.

The results yielded by this approach are summarized in Figure 4 (a) in the form of a frequency response plot of the coil inductance. Maxwell's analytical approximation for this inductance is 38.7 mH ; this value is also displayed in that plot as a horizontal line for comparison.

The same Flux3D project may be used to determine the frequency behavior of the Joules losses in the coil of Figure 3. The losses may be once again evaluated with a sensor, yielding the results shown in Figure 4 (b). The evolution of the losses density at increasing frequencies is shown in Figure 5.

Figure 4. Frequency behavior of the inductance (a) and of the Joule losses (b) evaluated in a Magnetic Steady State AC Flux3D application using a coil conductor region with losses and detailed geometrical description.

Figure 5. Evolution of the Joule losses distribution in a winding modeled with a coil conductor region with detailed geometrical description for the frequencies of 1 Hz (a), 50 Hz (b) and 500 kHz (c).