Utilizing Helicoidal and Translational Symmetries Together in 2-D Models of Twisted Litz Wire Strand Bundles

In this article, a helicoidally symmetric 2-D model of a twisted litz wire strand bundle is studied. Suitable foundations for compatibility tools are presented to use such a model as a part of a larger system which does not follow a helicoidal symmetry. A measure for the symmetricity of a field is proposed. This provides the foundations for the coupling of helicoidally symmetric and translationally symmetric models with good understanding of the introduced approximation error.


I. INTRODUCTION
I NCREASING operating frequencies of power electronics requires using litz wires to keep eddy-current losses of windings at an acceptable level. Important applications for litz wires are, e.g., wireless power transfer coils [1] and windings in power electronic converters' magnetic components [2] and high-speed electrical machines [3].
Litz wires are made of thin electrically isolated strands of conducting material. Typically, there are hundreds or thousands of strands assembled in a bundled configuration. The strand bundles and bundles of bundles are usually twisted in a helicoidal shape to even out the proximity effects caused by the adjacent bundles or winding turns.
Windings made of litz wire are difficult to model due to their complex and fine-detailed multi-scale nature. However, detailed modeling of litz wires is crucial, as existing research suggests that choosing a suitable litz wire configuration is more complex than picking the strand diameter and the number of strands. It is reported in [3] that, e.g., the bundle configuration and shape of the strand have a significant impact on the total losses of litz wire windings in high-speed machines.
The computational burden of simulating the losses emerging in a device which includes litz wire windings is prohibitive without the application of special techniques. Because of this, the engineering field is constantly looking for new methods of incorporating litz wires more accurately in the computational models [4], [5], [6]. Symmetries of the modeled device, e.g., translational symmetry or rotational symmetry, are typically exploited to reduce the problem domain from 3-D to 2-D. If neither symmetry is directly applicable, the so-called multislice models can be used [7]. This reduction in dimension typically sacrifices some of the 3-D properties of the problem, one of which is the helicoidal twisting of the strands and bundles. To account for the twisting in an approximate manner, one needs to use multiple slices where the locations of the strands are varied from slice to slice. Manuscript  In this article, a 2-D model for a twisted strand bundle is developed using helicoidal symmetry and dimensional reduction [8]. This method has been used successfully in computing self-fields of superconducting cables [9] and power cables [10], but there was no attempt to connect the helicoidally symmetric model to any larger system. The novelty of this article is in using the familiar concept of harmonic fields, commonly used to model time-periodic signals, to capture the twisting effects.

II. METHODS
A 2-D model of a litz wire with n str parallel strands is considered. The wire is assumed to be helicoidal with a pitch length l z . The computational domain is divided into three parts: the Cartesian domain c , the twisting effects' domain tw , and the helicoidal domain h (see Fig. 1). In a cross section of a litz wire, the strands are more distorted the further they are from the twisting axis. We use circular strand cross sections for simplicity. The 3-D geometry is extruded from the 2-D slice, and hence the 2-D geometry is an accurate representation of the 3-D geometry. The 3-D geometry, however, is only an approximation of a real litz wire structure.
Our aim is to keep tw as small as possible and free of conducting and nonlinear materials. In this article, we do not yet have the tools to incorporate c into the computational model. Hence, we consider the area = tw ∪ h . Two charts are used to cover the domain: a Cartesian chart χ c which covers the whole domain c ∪ and a helicoidal chart χ h which covers (see Fig. 2). As the charts are overlapping in , a transition map χ = χ −1 h • χ c exists and we can change the representation of fields in between the coordinate systems. It might be confusing that χ c and χ h are not explicitly stated. What we have is certain properties, such as material parameters, measured using χ c and we want to represent them using χ h . That is why we only need to instantiate the transition map χ which defines the relationship between χ c and χ h .
For brevity, let c = cos(αw) and s = sin(αw), where α = (2π/l z ). The transition map is χ :  The Jacobian matrix of the transition map is The magnetostatic problem is defined by the equations where d is the exterior derivative, H is the magnetic field strength, B is the magnetic flux density, J is the current density, and ν is the reluctivity tensor. By representing the magnetostatic problem in the (u, v, w) coordinates, we can reduce the problem into a 2-D translational symmetric problem. See [8] and [9] for a detailed description of the reduction.
The difference compared with a Cartesian translational symmetric problem is visible in the material law ⎡ The translational symmetric problem is formulated by having the out-of-plane components of the vector potential A w and the magnetic field H w as unknowns. Equations (3)-(5) then become The system consisting of (7) and (9) is discretized with 2-D finite-element discretization and solved in the w = 0 plane.

III. SYMMETRY OF A FIELD
We interpret symmetry of a field as immutability of the field in the direction of the symmetry. Formally, as a measure for symmetricity we propose the Lie derivative of a field along the symmetry flow. For example, the Cartesian translational symmetry flow is expressed using a vector field Z , such that it defines the z-direction at every point in the domain. The vector field Z is part of the x yz-coordinate frame. It acts as a directional derivative for the coordinate 0-forms x, y, and z such that Z (x) = Z (y) = 0, and Z (z) = 1. The Lie derivative of any p-form η with respect to vector field Z can be calculated using Cartan's magic formula where i Z is the contraction along the vector field Z . The contraction is linear with respect to its vector field argument, and hence i Z can be expressed in the uvw-coordinates as where U , V , and W are the vector fields along the coordinate axes of the helicoidal uvw coordinate system. Note that because z = w in the transition map, dz = dw holds, but Z = W does not. The expression for the Lie derivative of a helicoidally symmetric H can be derived to be In (12), we have used the fact following from our interpretation of symmetry that for a helicoidally symmetric H , the partial derivatives with respect to coordinate w of its component Based on our interpretation of symmetry, the field H is symmetric with respect to a vector field Z if Our hypothesis is that the field H , emerging from the helicoidally twisted wire, can be considered z-symmetric when L Z H becomes negligible. If the pointwise norm of the Lie derivative ||L Z H || is small enough around a point, H can be considered translationally symmetric, with respect to Z , around that point. Fig. 3 visualizes how the symmetricity of the field varies. The norm has the highest values close to the bundle, and the value decays with respect to the distance from the bundle. The decay does not happen quickly enough to justify neglecting the twisting effects right outside the bundle. In a winding made of twisted litz wires the bundles are packed close to each other, and thus the twisting effects penetrate the neighboring bundles and any, possibly magnetic, nonlinear and conducting, materials close to the windings. Therefore, by simplifying a litz bundle to be translationally symmetric with respect to the z-coordinate, we introduce error to the computational model.

IV. MULTIHARMONIC FIELDS
Since bundle twisting effects on the magnetic field distribution cannot be neglected, we need a way to incorporate them to the Cartesian translational symmetric model in c . We would like to express a helicoidally symmetric field with something that is more compatible with the z-symmetry. As helicoidally symmetric fields are periodic with respect to the z-coordinate, they can be expressed using a Fourier series. We are interested in how well such fields can be approximated using a finite number of harmonics, i.e., what we call z-multiharmonic fields.
Analogously to the multiharmonic models in the time domain [11], we say that the field H is z-multiharmonic if there exist complex valued 1-formsĤ k such that To compute the coefficientsĤ k , the representation of the helicoidally symmetric H , obtained in the uvw-coordinates, must be changed. We can change the representation of H , inside tw , with the help of the Jacobian as ⎡ The coefficientsĤ x,k ,Ĥ y,k , andĤ z,k can then be calculated as integrals over one period of length l z along the z-directional fiber of the Cartesian coordinate system where i ∈ {x, y, z}. Since the solution is helicoidally symmetric, the values of the field on a z-directional line are equivalent to values of the field in a circle around the bundle on the  x y-plane. Hence, the coefficients can be computed from the information available in the 2-D helicoidally symmetric model.

V. RESULTS AND DISCUSSION
Wires with three and five copper strands of diameter 0.1 mm, of pitch length 1 mm, and of net current 1 A were simulated using the 2-D symmetric model and a full 3-D model. The parts outside the strands are considered as air. In Fig. 4, the z-component of the magnetic field is shown. The field is the strongest in the middle and decays rapidly outside the bundle. This component of H is neglected entirely in a Cartesian translationally symmetric 2-D model. In Fig. 5, the current density in the strands is shown. It is worth to note that as a simplification in the 2-D model, J is forced to be parallel to the helicoidal strands. In the 3-D model used for comparison, this restriction is not present. The 3-D model was exited by a voltage source V in , whereas the 2-D model was exited by an equivalent w-directional electric field E in = (V in /l z ). Both the inputs resulted in a net current of 1 A in the wire. The results between the 2-D and 3-D models match well, which corresponds to the observations made in [9].
Next, the harmonic components of the magnetic field were computed with n str = 2, 3, . . . , 9 copper strands in the bundle. The strand diameter, pitch length, and net current are 0.1 mm, 35 mm, and 1 A, respectively. Fig. 6 visualizes the harmonic componentsĤ k of H for the case n str = 3. We observed that Four biggest harmonic components of H by magnitude in the xy-plane in the case of three strands. The colormap shows the norm ||Ĥ k ||, and the quiver shows the real part Re{Ĥ k } of the component field. Fig. 7.
Behavior of the four biggest harmonic components of H by magnitude, with respect to the distance from the center of the bundle in the case of three strands. the four most significant harmonic components, in descending order by magnitude, are k = 0, k = n str , k = 2n str , and k = 3n str . The k = 0 component is analogous to a dc-offset in a time-multiharmonic signal. It is the component of the helicoidally symmetric H which already is z-symmetric. The components k > 0 correspond to the nonsymmetric parts which would be completely neglected in a translationally symmetric model. From Fig. 7, it can be seen more precisely how the different harmonic components behave with respect to the distance from the center of the bundle. The k = 0 harmonic decays inversely proportional to the distance, as is expected based on the Biot-Savart law. The k > 0 components decay more rapidly. This matches our intuition that one cannot differentiate a twisted and an untwisted wire bundle if they are observed at distance.

VI. CONCLUSION
The behavior of the twisting effects of a litz wire bundle was studied, using a helicoidally symmetric 2-D model.
In this article, the harmonic components are computed in a postprocessing phase, so the coupling between the Cartesian translational symmetry and helicoidal symmetry is only unidirectional. A bidirectional coupling is required to fully use the approach. It is our hypothesis that any helicoidally symmetric field in h ∪ tw can be represented as a z-multiharmonic field in c , and any z-symmetric field in c ∪ tw can be represented as a w-multiharmonic field in h . Then, by introducing suitable coupling conditions in tw , we can leave the harmonic coefficients as unknowns and solve them from a single equation system. The tools and observations presented in this article serve as a foundation for future research on this topic.