Air Gap and Edge Effect in the 2-D/1-D Method With the Magnetic Vector Potential ${A}$ Using MSFEM

Abstract

Eddy currents (ECs) are simulated in a single laminate, representing the whole core of an electrical machine. Despite this drastic reduction in the complexity of the problem, a 3-D finite-element model turns out to be still too expensive for simulations. To overcome this difficulty, 2-D/1-D methods are used. This article presents a solution to consider both air gap and edge effect (EE) based on the multiscale finite-element method (MSFEM) using the magnetic vector potential (MVP) A. Linear material properties are assumed; therefore, this article is carried out in the frequency domain. The new 2-D/1-D MSFEM is discussed, and various simulation results are presented. Index Terms-2-D/1-D method, eddy-current (EC) problems, edge effect (EE), iron core, lamination, magnetic vector potential (MVP) A, multiscale finite-element method (MSFEM).

Multiscale Finite Element Formulations for 2D/1D Problems

“…The EE is represented by φ 1 grad u 1 and by φ 1,z (0, 0, w 1 ) T in (20) and (21), respectively. Laminar currents due to φ 0 1 grad u 10 generate a total magnetic flux (26), which is perturbed either by (27), see [11], or by (28). The zcomponent of curl(φ 1 A 1 ) provides a smoothing of the magnetic field at the transition from the iron sheet to the air.…”

“…and known steps considering the BCs (30) to (44) are carried out for the FEM ([10], [11], [12]). Finite element subspaces of the potentials have been selected as follows: T 0 ∈ H(curl, Ω), T 2 and A 1 ∈ H(curl, Ω m ), Φ 0 and u 10 ∈ H 1 (Ω) and u 1 and w 1 ∈ H 1 (Ω m ), see [13].…”

“…A 2D/1D MSFEM using trigonometric functions across the thickness of the sheet can be found in [9]. Our methods also have to consider eddy currents including the EE, see [10] and [11], account for an insulation layer in between the iron sheets, facilitate boundary conditions (BCs) to exploit planes of symmetry, and use Biot-Savart-fields (BSF) to avoid modeling of conductors carrying known currents. So far, an excitation has been introduced only by proper BCs in the tiny problem in [10] and [11].…”

A Hierarchical Error Estimator for the MSFEM for the Eddy Current Problem in 3-D

“…with φ1 := φ 0 dz and φ3 := φ 2 dz (13) and the unknowns γ 0 , γ 2 ∈ H 1 (Ω 2D ) and γ 1 , γ 3 ∈ H(curl 2D , Ω 2D ) to be determined. Note that the estimator on the right hand side of (9) consists of σγ, which is equal to zero in the insulation because of the conductivity, and curl T 2D/1D , which has only components containing the shape function φ 2 , see (7), which is also zero in the insulation. Therefore it suffices to consider only the domain of the conducting material for the construction γ, i.e.…”

“…This enables the solution of the problem within a single iteration while requiring only a mesh for the two dimensional cross section of the sheet. It is also able to include the insulation layers between sheets and correctly treat the edge effect [7].…”