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Efficient Finite Element Computation ofCirculating Currents in Thin Parallel Strands
Antti Lehikoinen and Antero ArkkioAalto UniversityDept. of Electrical Engineering and AutomationP.O. Box 13000, FI-00076 Espoo, Finlandemail: [email protected]
1 IntroductionElectrical machines often utilize stranded parallel conductors to reduce the skin-effect losses. This practice can leadto uneven total current distribution among the strands, increasing the resistive losses. Direct finite element analysisof circulating current problems can be computationally costly due to the large number of nodal unknowns in theconductor mesh. Two efficient finite element formulations are proposed to solve circulating current problems witharbitrary winding configurations.
(a) Mesh we’d like to use. (b) Mesh we’ve been forced to use until now.
Figure 1: Meshing the strands results in a large number of nodal potentials.
2 TheoryApplying the Galerkin approach to the Aϕ-formulation of an eddy-current circuit problem yields the block matrixequation SAA+M ∂
∂t SAu 0MuA ∂
∂t −I Rui
0 Riu Rii
A
ui
=
00U
, (1)
with the following block elements
[M]r,c =∫Dl
σφrφc dDl,[SAu]
r,l =∫Dl
− 1leσφr dDl
[MuA]
l,c = Rl
∫Dl
σφc dDl (2)
directly related to the strand l.
3 Proposed methods1) In the point-strand method, the strands are assumed very thin. Thus, (2) can be reduced to
[M]r,c ≈le
Rlφr(xl)φc(xl)[
SAu]r,l ≈− 1
Rlφr(xl) (3)[
MuA]l,c ≈ leφc(xl)
where xl is the strand center.2) In the polygon-strand method, strands are approximated with polygons independently from the mesh used. Integralsin (2) are evaluated exactly with Gaussian quadratures on a background mesh.
[M]r,c =∫Dl
σφrφc dDl (4)
= ∑e∈Dl
∑k∈e
∑i
wiφ̂rφ̂c
(F−1
e Fbge,k (x̂i)
)|detJ
(Fbg
e,k (x̂i))|
Figure 2: Example of the background mesh.
4 Simulation resultsSynchronization of a simplified four-pole PM machine was simulated with the proposed methods. The brute-forceapproach was used as a reference.
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Width (mm)
Hei
ght (
mm
)
Figure 3: Cross-section of the machine.
Three sets of results can be seen below.
6 6.5 7 7.5 8 8.5 9 9.5 10
40
60
80Open slots; unrefined mesh
6 6.5 7 7.5 8 8.5 9 9.5 10
40
60
Open slots; refined mesh
6 6.5 7 7.5 8 8.5 9 9.5 10
20
30
40
Time (ms)
Semi−closed slots
Pcc
(W
)
Figure 4: Circulating current losses by the proposed methods (red, black) and the brute-force ap-proach (blue).
For an open-slotted machine, the proposed methods yield reasonably accurate results after one uniform mesh refine-ment in the slot region. With semi-closed slots, a close agreement is reached with the initial coarse mesh.As can be seen in the Table below, the proposed methods yield significant savings in computation time even with therefined mesh.
Table 1: Simulation Details.No. of nodes Computation time (s)
Brute-force 5288 1052.7Point-strand (unrefined)
442129.5
Polygon-strand (unrefined) 124.7Point-strand (refined)
938153.2
Polygon-strand (refined) 158.3
5 Conclusion
• Two methods proposed for FE analysis of circulating currents.• Arbitrarily coarse meshes can be used.• Reasonably accurate results at a fractional computational cost.
