Efficient FE Modelling of CirculatingCurrents in Stranded WindingsAntti LehikoinenAalto University School of Electrical Engineeringantti.lehikoinen@aalto.fi

September 2, 2015

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Problem Background

I Skin effect losses in largeconductors→ split it into strands &connect in parallel

I Different flux linkage for eachstrand→ different current in each strand→ called circulating currents

I Losses

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A Primer on Finite Element Analysis

I (One) Idea: solve Maxwell’s equations by writing

B (x) = ∇× A (x) (1)

A (x) ≈N∑

k=1

akφk (x) (2)

and "minimizing"∇× µB − J = 0 (3)

→ matrix equation with entries of type

Sij =

∫x

f(x, φi , φj

)dx (4)

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Mesh

I Problem domain divided intoelements (meshing)

I φ and∫x

f(x, φi , φj

)dx calculated

based on the meshI Problematic with stranded

conductorsI dense meshI lots of elementsI lots of ak

I Mesh like this not required

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Solution

I Use a coarser non-conformingmesh

I Approximate strands with Diracdelta functions →∫

x

f(x, φi , φj

)dx ∼ f

(x0, φi , φj

)(5)

where x0 = strand centers→ significant reduction incomputation time

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ExampleI Time-stepping simulation of two PM motors

I Open and semi-closed slots – otherwise identicalI 48 strands per slot

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

)

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Circulating current lossesI Circulating current losses as a function of time

6 6.5 7 7.5 8 8.5 9 9.5 10

30

40

50

60

Open slots

Brute−force

Proposed

6 6.5 7 7.5 8 8.5 9 9.5 10

20

30

40

Time (ms)

Semi−closed slotsPcc

(W

)

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Computation times

I Proposed method ∼ 7 times fasterNo. of nodes Computation time (s)

Brute-force 5288 1053.7Proposed 938 148.7

I Real machines often have ∼ 300 strands per slot→ simulation now possible

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Conclusion

I A method was proposed for FE analysis of circulatingcurrents

I Applicable for any circuit configuration and FE formulation

I Significant computational savings even in simplifiedproblems

I Simulating realistic machines now feasible

I Questions?