Calculation of AC losses in current forced conductors using 3D finite ...

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 2, MARCH 1990

CALCULATION OF AC LOSSES IN CURRENT FORCED CONDUCTORS USING 3D FINITE ELEMENTS AND THE A$V METHOD P.J.Leonard

D.Rodger University of Bath, Claverton Down, Bath BA2 7AY, UK

Abstract

The term within {} is added as a penalty term constraining the divergence of A to be zero [Z]. To ensure that the solution is unique we add the additional constraint :

A scheme for modelling the current distribution in massive voltage forced or current forced conductors. The method is based on a 3D Finite Element model which uses the A$V field representation. The scheme has been verified by simulating a toroidal choke and comparing the predicted a.c losses with experimental measurements.

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on the A+ interface. In nonconductors we solve for the magnetic scalar potential :

Introduction

This paper presents a scheme for modelling currents in massive conductors, taking external sources as well as induced eddy currents into account. There is a need for this approach when proximity or skin effects are significant. In such a case the distribution of current can not be assumed and traditional methods would be difficult to use. This is particularly true in the case of a current forced transient situation. The scheme is based on the A$V formulation with the gauge of V.A = 0 implied by a penalty term. This leads naturally to a model in which both voltage and current sources can be imposed on the boundary of the finite element model.

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R.J.Hil1- Cottingham

The interfaces between conducting and non-conducting regions are coupled via the continuity of normal flux and tangential m.m.f. [I] The magnetic scalar region also requires a cutting surface for each current or voltage forced circuit [3] .

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The Finit e Element Approximat ion

The finite element method uses a set of basis functions. These nodal shape functions form a set w; from which the field is approximated by the unknown nodal potentials :

The A$V Field Representation

The classical method of representing the electromagnetic field in terms of a magnetic vector potential and an electric scalar is used within the conductor :

8A E=---VV

The Galerkin procedure yields a set of equations by substituting the field approximation into the governing equations and ensuring that the error is orthogonal to the basis functions. After the usual transformations we obtain, in nonconductors :

at

In non-conductors it is more economic to use the magnetic scalar potential II, [I] : In conductors : Substituting these representations into Maxwell's equations yields, in conductors :

=

v . u ( g + v v )= o

(5)

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aVw;

OAv

0018-9464/90/0300-0490$01.OO 0 1990 IEEE

(g+

J

wiH x d r (12)

'AV

VV) dR =

I

'AV

w;J .d r

(13)

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Voltage and current forced condi- 5 t ions

Typically the voltage or total current may be prescribed on one or more conductors on the bounday of the finite element model, shown in figure 1.

Results

The scheme has been used to model a toroidal choke. The choke Was constructed by winding two layers of 26 turns of wire ~ o u n d a cylindrical former. The resulting double layer solenoid was then stretched around a second former and pairs of turns tied together. This type of construction is used when a. very low leakage field is required. A simplified single layer version of the coil is depicted in figure 2

Figure 2: Simplified coil showing the geometry

Table 1 Choke specifications Figure 1: Typical Finite Element Model wire diameter

The voltage forced condition can be modelled by substituting the known voltage into equation 13. The current forced case requires some additional work. If the current distribution on the problem boundary was known then it would be a simple matter to introduce the appropriate source term into the r.h.s. of equation 13. However in practice it is usually the total current entering over a particular surface that is given. If we assume that the current flow is normal to the face then it follows that the voltage is constant over the face. We can thus replace the set of unknown voltages over the face with a single unknown, V, .

2.65mm

The computer model assumed that the current could be reasonably approximated by a set of 26 pairs of current loops allowing 1/104 of the device to be simulated. Figure 3 shows the 2 half turns embedded in a wedge of air.

The basis function for this unknown is the sum of all the nodal basis functions lying on that face. If i spans only the nodes lying on the face : w, =

cui

The total current flowing into this face can be prescribed by setting the r.h.s. term for the equation corresponding to U,. Figure 3: Segment used for computer model

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The mesh used to model the coil was constructed by expanding a two dimensional base plane into three dimensions and adjusting the nodal co-ordinates to give the required geometry. Figure 4 shows the resulting mesh.

Figure 6: Current distribution at zero current lOkHz

Figure 4: Finite element mesh

The magnetic scalar potential was cut over the surface spanning the conductor circuits giving two surfaces on which 11, is constant but unknown. The model was current forced by the method described above, in this case two sources were required one for each loop. The device was simulated at 3 frequencies, the resistance was calculated by prescribing a nominal input current to each half turn and measuring the voltage drop. Two simulations were performed, one using first order iso-parametric bricks and a second with the outer layer of conducting elements modelled using second order hierarchical elements allowing the skin depth to be modelled better . The results are shown Table 2.

Figure 7: Contours of magnetic scalar potential at peak current

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Conclusions

The A+V method has been shown to give reasonable results when applied to the toroidal choke problem, considering the double helix was idealised as a set of circular conductors. Selectively increasing the order of representation has been shown to be a good technique for modelling skin depth. One disadvantage of the scheme is the number of variables per node, 5 on the conductor to non-conductor interface. This leads to a relatively dense matrix which is costly to solve. Figures 5 and 6 show the current distribution for the lOkHz case at peak current and zero current. The contours of magnetic scalar potential are shown in figure 7.

References [l] R.D Pilsbury : ’A three-dimensional eddy current formulation using two potentials: the magnetic vector and total magnetic scalar potential’, IEEE Trans Mag. Vol 19. No. 6, pp2284-2287 ,1983

[2] W. Renhart, H. Stogner and K. Preis: ’Calculation of 3D eddy current problems by finite element method using either an electric or a magnetic vector potential.’, IEEE Trans. Mag., Vol 24, No. 1, August 1987

[3] P.J.Leonard and D.Rodger : “ A new method for cutting the magnetic scalar potential in multiply connected eddy current problems” IEEE Trans. Mag. Washington ( To be published sept. 1989 ) Figure 5: Current distribution at peak current lOkHz