1122 IEEE TRANSACTIONS ON MAGh'ETICS, VOL. 28, N0.2, MARCH 1992

The Modelling o f Segmented Laminations in Three Dimensional Eddy Current Calculations

B.C. Mecrow, A . G . Jack

Department of Electrical and Electronic Engineering The University of Newcastle Upon Tyne Newcastle Upon Tyne, NE1 7RU, U.K.

Abstract - Segmented cores commonly occur in large transformers and turbine generators where the physical dimensions of the core exceed the maximum available single lamination size. This paper describes a method for explicit three dimensional modelling of eddy currents in such laminated iron cores, where the core is split into a number of overlapping segments. The fields are calculated using the T - 0 method, with multiple T vectors to represent the layers of overlapping laminations.

I. INTRODUCTION

The magnetic flux paths of virtually all electromechanical devices are laminated to inhibit induced eddy current flow. However, fringing and endwinding effects result in a component of flux which travels normal to the plane of the laminations thereby inducing eddy currents in the plane of the laminations. A considerable amount of work has been directed towards calculation of these skin limited eddy currents and the resultant loss distribution, with the authors developing three dimensional finite element codes for the purpose [l]. In large devices it is necessary for the laminations to be made up of a number of segments, with overlapping between successive layers for increased mechanical integrity. This segmentation and overlapping has an important influence upon the resultant eddy current distribution.

Phemister and Wymer [ 2 ] presented an analytical formulation in which they determined the two dimensional field distribution in segmented overlapping rectangular laminations, resulting from a specified field source normal to the laminations. More recently Preston and Timothy [ 3 ] used a three dimensional model of a small core section, with explicit modelling of each lamination. Both approaches yielded very useful information, but are not suitable for inclusion in a model of a complete turbogenerator end-region simulation - an essential requirement if accurate stator core loss densities are to be calculated. Within this paper a multiple electric vector potential formulation is developed to directly include the segmentation effect in a three dimensional eddy current model, without the need to model individual laminations.

11. NUMERICAL FORMULATION

If a segmented core has no overlapping between successive layers of core plate then eddy currents set ,up within each segment will force flux travelling normal to the plane of the laminations into a skin around the periphery of the segment, and standard solution methods for laminar eddy current fields are appropriate. If, as more typically occurs, successive layers are displaced so that the centre of a segment in one layer coincides with the the edge of a segment in

next then the situation is far more complex. It must be considered whether the normal component of magnetic flux redistributes across each successive layer of laminations or only responds to the average effect of the induced eddy currents in the different layers.

By consideration of the geometry of the problem it is valid to make the assumption that magnetic flux travelling normal to the plane of the laminations does not redistribute itself on a lamination by lamination basis because of the lapping effects. This is not to say that the segmentation and lapping has no overall effect upon the flux, but rather that the influence of the different segment positions is a bulk effect only.' Such an assumption is valid because of the very large aspect ratio of laminations i.e. their thickness is typically less than 0.1% of the segment width. For the magnetic flux to remain near the edges of each successive segment the path length taken would have to be, approximately one thousand times larger with a much smaller cross-sectional area. Magnetic saturation alone would preclude such an eventuality, consequently it is not necessary to model each lamination individually, but instead only to have a valid representation of the eddy currents in the repeating layers of the segments.

The formulation is best illustrated with an example. Fig. 1 shows part of a turbogenerator core with four stator teeth per segment, lapped so that the segmentation pattern repeats every other layer of laminations. Let the first, third, fifth, etc. segment layers be circumferential pattern A and the second, fourth, sixth, etc. layers be circumferential pattern B. The eddy currents induced in pattern A may be denoted by JA, and those in pattern B by JB. Hence

V x TA = JA

V x TB = JB and

where TA and TB are separate potential describing' functions directed normal to the plane of the laminations. Therefore, although TA and TB are vectors they each have only a single component. Had the lapping been such that the geometrical pattern repeated every three lamination layers then a third T vector would need to be introduced.

From the assumption that the magnetic flux does not redistribute between laminations then

and

The half arises because JA and JB are each present over only half the volume of core., From eqns.(1),(2) and (4)

Manuscript received July 7,1991, 0018-9464/92$03.00 0 1992 IEEE

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Pattern A Pattern B (a) end view

(b) Plane view

Fig.1 A segmented turbogenerator stator core with lapping between adjacent layers of laminations.

half

H = 1 ( T ~ + T ~ ) - vn 2

(5)

where 0 is a magnetic scalar potential source. Thus the divergence of the magnetic fieid may be written as

v.[ g ( ~ ~ + TB) - pvn] = o (6) 2

As the magnetic field source for JA and JB is the same then the two separate electric fields generated by these currents both have the same curl. i.e.

Thus substitution of eqns.(l), (2) and (5) into eqn.(7) results in

v x 1 v x T~ - - a [E(T~ + TB) - vn] (8) D dt 2

and v x 1 v x T~ - - a [&(T~ + TB) - vn] ( 9 )

D dt 2 where U is the electrical conductivity of the laminations. Because of the laminar nature of the eddy currents the vectors TA and TB can be constrained to be directed normal to the plane of the laminations, and’ hence three variables must be solved for: TA, TB and n. This is done by satisfying the divergence eqn.(6) and the components of the vector eqns. (8 ) and (9) normal to the planar flow of the eddy currents. Employing the method of weighted residuals these equations can be derived in integral form to produce a symmetrical set, employing complex arithmetic to denote the harmonio nature of the time variation of all fields.

Eqn. ( 6 ) gives

Eqn. ( 8 ) gives

The complete eddy current formulation is embedded in a three dimensional finite element model of a turbogenerator end-region. MMF sources are employed to represent the stator winding currents and external to the stator core a magnetic scalar potential describes the magnetic field representation. In order to maintain acceptable nodal limits the circumferential boundaries are limited to cover the minimum arc over which the eddy current patterns repeat. Dirichlet nodes are used to impose the segment boundaries, with periodid boundaries linking TA and TB.

111. RESULTS

A . Simplified Problem

Within a turbogenerator axially directed magnetic flux within the stator core forms the source of eddy currents induced in the laminations. This axial flux is a result of stator and rotor end-winding currents and air-gap fringing flux in the machine end-region. Hence a complete solution to the eddy current problem requires a full turbogenerat0.r end-region model.

For testing purposes the formulation was initially applied to a simplified problem of a segmented turbogenerator core, half lapped with four stator teeth per segment. In order to simplify the problem the flux sourcing term, Vn normal to the plane of the laminations, was fixed and held constant across the complete cross-section of the lamination segments. In this case symmetrypermitted limitation of the model to two slot pitches: A solution was obtained for two cases; one without lapping between segments, and one with the half lapping.

Figure 2(a) illustrates the resultant eddy current pattern at one instant in time for the case where segmentation is included, but no account has been taken of the overlapping of segments in successive layers. It is clear that both the normal component of magnetic flux and the segment eddy currents flow exclusively in a skin around the entire periphery of the segment. The depth of this skin is a function of the electrical frequency, the stator core axial permeability (dictated by the gaps between laminations) and the lamination electrical conductivity and is typically about 20mm.

Figure 2(b) illustrates the equivalent eddy current pattern at the same instant in time with overlapping segmentat,ion included in the manner outlined in this paper. Within the stator teeth the eddy current flow pattern remains similar to before, but in the core back region the pattern is substantialLy different, emphasising the influence of the overlapping. Close examination of the results has revealed that in the region around the joints between segments there is less total core eddy current reaction. Some axial flux is therefore able to pass at this position, so that in this half lapped example there is not only increased axial flux at the circumferential edges of each lamination, but also at the lamination centre (corresponding to a joint in the next lamination layer). This latter component of flux forms a source for eddy currents which are evenly distributed throughout the stator core back, to some extent negating the skin limiting effect which previously occurred along the segment boundaries.

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(a) No lapping

(b) Half lapping

Fig.2 Eddy current distribution within a segmented stator core, illustrating the importance of lapping effects.

The eddy current contour plot of fig.2(b) shows a marked similarity to both the results of Phemister and Wymer [2] using analytical methods, and those of Preston and Timothy [3], modelling individual laminations.

B. Full End-Region Solution

Following the simpler test problem outlined above the general formulation was employed in a full turbogenerator end-region solution. The machine modelled is a, 660MW, 50Hz turbogenerator with three stator teeth per segment. This machine has third lapping within the end-region of the stator core (i.e. the segments in successive layers of laminations are displaced by one slot pitch in this case). The third lapping necessitated extension of the fozmulation to have three current describing vectors, one for each repeating layer, but -in all other respects the equations are as given in section I1 above.

A full three dimensional end-region solution requires very large computing resources, and therefore every effort has been made to minimise the size of the problem. Two major assumptions have been made in order to facilitate this: the first is to assume that within the stator teeth and core the problem is time periodic

over a single slot pitch - this is in fact true for third lapping in three tooth pitch segments, but low order spacial harmonics of magnetic flux occurring due to magnetic saturation and phase belt harmonics are neglected. The second assumption is that all the spatial slotting harmonics which arise in the stator teeth and core are insignificant in areas of the machine end region distant from the stator core itself. This latter assumption has permitted a quasi-three dimensional model of much of the machine end-region, which is then matched directly into a full three dimensional model of the stator core and its close vicinity. The second order three-dimensional meshes employed are shown in fig.3.

The above mesh and formulation has been used to determine the stator core lamination eddy current distribution for the open-circuit condition. Fig.4 shows the eddy currents in a lamination right at the core end at two positions, one in the direct axis of the machine (where the source magnetic field is) and one in the quadrature axis. Note that this machine has "Pistoye slots" in the stator teeth which break up the teeth eddy current pattern. The axial flux for the open-circuit condition is primarily air-gap fringing flux, which is almost totally in the stator teeth - it is therefore in this region that the eddy currents dominate, and the core back is essentially just a return path for these currents.

A s the axial flux travels further into the stator core it tends to redistribute itself more evenly across the lamination, and there is thus a corresponding shift in the segment eddy current pattern. Fig.5 shows the resultant eddy current pattern at such a position, with significant eddy currents evident in the core back. As with the test case earlier the eddy currents are confined to a skin at the inner and outer boundaries of

(a) Quasi-3D end-region (b) 3D stator core mesh mesh

Fig.3 Finite Element meshes used for turbogenerator field calculation. Boundaries betwen meshes are directly linked in the formulation.

the stator core segment, but are not skin limited around the radial joints because of the lapping. It is also interesting to note how there are no significant core back radial currents in centre third of the coreplate segment because of the third lapping.

The overall effect of segmentation and lapping upon eddy current losses is of particular interest. The segmentation will obviously increase the overall resistance of the eddy current path; therefor? if the problem is flux forced there will be a V /R type situation and the losses will fall; conversely if there is current reflection within the stator core an 12R situation will arise and the losses will fall. In practice the problem is somewhere between the two; within the stator core back the latter exists, but wlthin the stator teeth the eddy currents are resistance limited and the former situation arises. The results obtained from this work indicate that for the open-circuit condition the losses actually reduce by over 30% due to the presence of segmentation and third lapping.

The results may appear to be in direct conflict with those of Preston and Timothy, who indicate an increase in loss due to segmentation, but it is considered that the difference is due to the type of boundary conditions employed. For the open-circuit condition most of the loss results from resistance limited eddy currents in the stator teeth, and therefore according to the above arguments a reduction in total loss may be i’expected. For the short-circuit condition much of the loss occurs in the core back and the opposite situation is likely to arise. Finally, the combined effect on load is almost certain to fall between these two cases.

IV. CONCLUSIONS

A formulation has been presented for the inclusion of segmentation effects in a three dimensional eddy current model of large laminated iron cores. The model has been developed for the study of eddy currents induced in the end laminations of turbogenerator stator Cores. It includes magnetic nonlinearity [I] and represents the overlapping pattern which occurs between successive lamination layers by introducing multiple electric vector potentials directed normal to the laminations. Such a formulation has permitted study of the segmentation effect without explicitly modelling each lamination layer - an impractical proposition in terms of the discretisation required. Results have demonstrated that the overlapping of successive segment layers has a very substantial influence upon the eddy current distribution in the stator core back. For the open-circuit condition a reduction in total core loss of over 30% is calculated to occur due to the presence of segmentation and third lapping.

ACKNOWLEDGMENTS

The authors would like to help NE1 Parsons Ltd. for both technical and financial assistance with this research.

REFERENCES

1. A.G. Jack and B.C. Mecrow, “The calculation of three dimensional electromagnetic fields involving laminar eddy currents”, IEE Proc. A , September 1987.

2 . T.G. Phemister and C. Wymer, “The effects of gaps between laminations on axial flux and eddy currents in an idealised stator core”, COMPUMAG, 1978, section 3.6.

3 . T.W. Preston and M.A. Timothy, “Eddy current distribution in segmented stator core laminations”, Universities Power Engineering Conference, 1990, pp631- 6 3 4 .

(a) Direct axis

, (b) Quadrature axis

Fig.4 Eddy current distribution in the end lamination of a third lapped core for the open-circuit condition.

Fig.5 Stator core eddy current distribution in a lamination several centimetres from the stator core end of a third lapped core for the open-circuit condition. Manuscript received July 7, 1991.