# Coil entities in symmetrical or periodical domains

## Introduction

Geometrical symmetries and periodicities are common constructive features in electromagnetic devices. Given their importance, Flux provides the user with a comprehensive set of tools dedicated to their proper and efficient representation in a project.

This documentation topic discusses the use of coil entities together with symmetries and periodicities in a Flux project. The subjects addressed in the next sections are the following:
• A reminder about the concepts of symmetry and periodicity in Flux;
• How to set the field Symmetries and Periodicites: number of conductors in series or in parallel during the creation of a coil entity;
• The effect of combining multiple symmetries in the definition of a coil entity in Flux.

## Symmetries and periodicities in Flux

Two tools exist within Flux to simplify the representation of symmetrical and periodical devices:

• The user may assign a Symmetry entity to the domain, both in 2D and in 3D applications. This tool is tailored for representing devices possessing a bilateral symmetry about a plane.
• The user may also create a Periodicity entity both in 2D and in 3D applications. This entity is well adapted to the description of devices whose geometries exhibit repetitive patterns, obtained either by translation or by rotation about an axis.
Figure 1 displays two examples of devices containing coils modeled in Flux with symmetry and periodicity entities.

Creating symmetries and periodicities in Flux projects is highly advisable. The user should include them in his or her project whenever possible because:

• they lead to significant simplifications in the project description since only a part of the device is represented. Consequently, shorter computation times and smaller memory requirements are verified;
• taking symmetries and periodicities into account in the project description leads to more accurate results. This assertion follows from the fact that the computed solution is not affected by the uneven meshing of the symmetrical and periodical parts, which is unavoidable in the case of a complete geometrical description.

From a finite element method point of view, creating symmetries or periodicities in a Flux project is equivalent to enforcing a set of boundary conditions to ensure the uniqueness of the solution of a boundary value problem. Of course, the imposed conditions should be coherent with the physics of the modeled device. Several types of boundary conditions are available through symmetry and periodicity entities (tangent or normal field condition and cylclic or anti-cyclic condition respectively) in Flux.

Nevertheless, in certain cases, the user may prefer to fully represent his device in the project (i.e., without the aid of Flux’ symmetry or periodicity entities). In such circumstances, it should be remarked that Flux Modeler (Flux built-in CAD module) can alternatively perform symmetry and repetition transformations on geometrical objects. These transformations are useful for building complex geometries with repetitive patterns and must be performed before the physical description .

## Associating coil entities to their periodical or symmetrical images

To successfully create a coil entity (i.e., a meshed coil region or a non-meshed coil magnetic source) in Flux, the user must specify how the "images" (or " copies") of the coil generated by existing symmetries and periodicities are connected to each other and to the associated FE coupling component in the external circuit.

Several possibilities exist, depending on the number and type of symmetries or periodicities existing in the project: in any case, this task is accomplished by setting the field Symmetries and Periodicites: number of conductors in series or in parallel of the coil entity to one of the following options:

• All in parallel;
• All in series;
• Number of symmetrical and periodical conductors in parallel, followed by an integer number provided by the user.
The example provided in Figure 2 illustrates the different available possibilities for the case of a coil conductor region in Flux 2D. The coil represented by the FE coupling component in Figure 2(a) is distributed to create a 6-pole winding, which is fed by the current source displayed in the external circuit. A periodicity (rotation about the Z axis, number of repetitions = 6, odd periodicity/anticyclic boundary condition) was employed to simplify the description of the model, which is limited to one-sixth of the winding, as shown in Figure 2(b) . The full winding is represented in Figure 2(c): while the original coil conductor region is displayed in red, its images are represented in other colors.

As already stated, different associations linking the "original" coil conductor and its "images" are possible. Table 1 lists all the coherent associations for the example of Figure 2. Notice that the same color convention established in Figure 2(c) was used in the circuits of that table.

Table 1. Possible associations between periodical conductors in the example.
Symmetries and Periodicites: number of conductors in series or in parallel Equivalent circuit association between conductors
All in series or Number of symmetrical and periodical conductors in parallel = 1 Number of symmetrical and periodical conductors in parallel = 2 Number of symmetrical and periodical conductors in parallel = 3 All in parallel or Number of symmetrical and periodical conductors in parallel = 6 It should also be remarked that terminals A and B marked on the circuits of Table 1 correspond to the terminals of the FE coupling component shown in Figure 2(a). Consequently, while the FE coupling component is always traversed by the same current, the current actually flowing in the coil conductor region (and in each of its periodical images) depends on the connection scheme chosen by the user.

In the previous example, for a current I fed to the FE coupling component by the external circuit, the current flowing in the coil conductor region and in its images may be either ± I, ± I/2, ± I/3 or ± I/6, according to Table 1. As a result, the magnetic flux density distribution displayed in Figure 2(d) also depends on the connection scheme adopted by the user: the field intensities computed by Flux2D will also be scaled by the factors 1, 1/ 2, 1/3 or 1/6 affecting the current intensities, depending on the case.

## Strategies for representing coils in domains with combined symmetries

The example in Figure 3 illustrates how symmetries may be combined in Flux to simplify the modeling of electromagnetic devices. That figure shows a single coil modeled in Flux2D using four different approaches:
• (a) without the use of symmetries;
• (b) with a single normal field symmetry along the horizontal plane;
• (c) with a single tangent field symmetry condition along the vertical plane;
• (d) with a tangent field symmetry condition along the vertical plane and a normal field symmetry condition along the horizontal plane.

As already mentioned, the same coil may be represented in Flux by any of these strategies. Depending on the specific context, one of them may be more convenient or justifiable than the others. In any case, the user should keep in mind that the number of coil entities required and their internal settings may be slightly different due to the combination of symmetries in the project.

For instance, the number of coil conductor regions required to completely represent the winding of Figure 3 in Flux2D is affected by the existence of tangent field symmetries (representing the physical condition at the coil axis) as follows:

• two independent coil conductor regions are required (with opposite orientations) when no tangent field symmetries are present, as in cases (a) and (b) of Figure 3.
• only one coil conductor region is required when a tangent field symmetry is present, as in cases (c) and (d) of Figure 3.
On the other hand, the number of turns provided during the creation of a coil conductor region depends on the existence of normal field symmetries. If the real coil being modeled has N turns:
• the number of turns in the coil conductor regions should be set to N/2 if a normal field symmetry plane is present, as in cases (b) and (d) of Figure 3.
• the number of turns in the coil conductor regions should be set to N in the absence of a normal field symmetry plane, as in cases (a) and (c) of Figure 3.

Table 2 summarizes the discussion above.

Table 2. Impact of symmetries in the modeling strategies for representing an N-turn coil in Flux2D.
Full model Half Model - horizontal cut Half model - axial cut Quarter model
Symmetries No symmetries Normal field symmetry Tangent field symmetry Normal and tangent field symmetries
Input coil conductor region N turns N/2 turns N turns N/2 turns
Output coil conductor region (with opposite orientation) N turns N/2 turns not required not required