SCIENCE

3-dimensional eddy-current calculationby the network method

Formulation using magnetic scalar potential fornonconducting regions

M J . Balchin, B.Sc. (Eng.), Ph.D., D.I.C., A.C.G.I., and J.A.M. Davidson, B.Sc. (Eng.), Ph.D.

Indexing terms: Eddy currents, Electromagnetics, Magnetic fields

Abstract: A network model for the representation of the general 3-dimensional electromagnetic field hasbeen presented in previous publications. Solution of the network was performed by an all mesh variablemethod. A method is described in the paper for the solution of the linked network model in terms of ascalar potential in the magnetic part of the field, and the mesh currents in the electric part. The validity ofthe solution method is confirmed by a comparison of calculations with experimental results, for a full 3-dimensional problem. The new node-mesh method is shown to give rise to considerably less variables thanthe all mesh variable method for the solution of the linked-network problem. A comparison is also madebetween the number of variables required by the node-mesh method and that required by some other dif-ferential equation formulations of the 3-dimensional eddy-current problem.

List of principal symbols

C — capacitanceCM,CE,N,NN = connection matricesE, e = lamellar and solenoidal components of

voltageG = conductance/, i = lamellar and solenoidal components of

current/M, m

PPT

= magnetomotive force (MMF), lamellar andsolenoidal components, respectively

= permeance= time derivative, d/dt= electric vector potential= magnetic flux= magnetic scalar potential

1 Introduction

In previous publications [1,2] a network model for the rep-resentation of full 3-dimensional eddy-current problems wasdescribed. The model consisted of two physically separate,but interlinking networks. One network was used to representthe magnetic region and the other to represent the electricregion. Electromagnetic coupling existed between the two net-works and was defined in accordance with Faraday's andAmpere's Laws. The complete linked-network model wassolved by the mesh method; giving a solution in terms of theminimum independent set of loop fluxes in the magneticcircuit and loop currents in the electric circuit. This was foundto be a convenient and accurate solution method for thelinked-network field model.

In many 3-dimensional power frequency eddy-currentproblems much of the field region may be nonconducting.The magnetic circuit of the network model is then muchlarger than the electric circuit part and it is often very largeindeed. For 3-dimensional network problems the number ofvariables required to solve a given network by a mesh methodis normally considerably greater than that required for a nodepotential solution. It is therefore more economical, in termsof the number of variables, if the magnetic part of the linked-

Paper 2380A, first received 11th May and in revised form 25th October1982The authors are with the Department of Electrical Engineering, Univer-sity of Bath, Claverton Down, Bath BA2 7AY, England

network model is solved for a scalar magnetic potential.A method is described in this paper for solution of the

linked-network model in terms of a scalar potential in themagnetic circuit and the loop currents in the electric circuit.The validity of the solution method is confirmed by a com-parison of calculations with experimental results, for a full3-dimensional problem.

Some aspects of the new node-mesh variable and the allmesh variable [1,2] method are compared. A comparisonis also made of the number of variables required by the node-mesh method and that required by the differential equationtype methods, to solve the 3-dimensional eddy-currentproblem.

2 Solution of the linked-network model using both nodeand mesh variables

The linked-network model has exactly the same form as thatdeveloped in References 1 and 2. Two physically separatenetworks represent the magnetic and electric field regions.Elements of the network models are shown in Fig. 1. In the

oo

(K) -o

Fig. 1 Elements of the linked-network model

a Electric circuitb Magnetic circuit

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electric circuit the conduction current flows in conductanceelement G and the displacement current flows in the capaci-tance element C. The lamellar component of current is repre-sented by the source /. The source e is solenoidal and es isa known source. In the magnetic circuit the permeance Pcarries the total flux 0. The source m is solenoidal and ms isa known source. There is no lamellar component of fluxbecause there are no free magnetic poles. The solenoidalpotential source terms e and m in the circuits represent theelectromagnetic coupling between the two networks. Themagnetic circuit exists throughout the field region, and forpower frequency problems the electric circuit exists only inconductors. Where both circuits exist, they interlink, as shownin Fig. 2, for a rectangular subdivision, with each node of theelectric circuit body-centred in a volume of the magneticcircuit and vice versa.

Fig. 2 Region of the subdivided field

• Electric, • magnetic

The potential-flow relations for all branches of the typeshown in Fig. 1 are given by the matrix equations:

es =

M+m

(1)

(2)

In Reference 2, Ampere's law was used to relate the solenoidalMMF sources m in the magnetic circuit to a set of closed pathor loop currents ij in the electric circuit. This is expressedby eqn. 11 of Reference 2 which is reproduced as eqn. 3below:

Cjt{M + m) = (3)

This formally expresses Ampere's law for each of an indepen-dent set of closed paths in the magnetic circuit, when thecurrent flow in the linking electric circuit is represented bycurrents in an analogous set of closed-loop paths. CM is thebranch-mesh connection matrix for the magnetic circuit. Acolumn of CM defines the branches forming a closed path inthe magnetic circuit. The number of columns in CM is equalto the minimum independent set of closed paths in the com-plete magnetic network. These closed paths can be found bydefining a 'tree' in the network. A tree defines a set of branches(tree branches) that connect all nodes in the network, but donot form any closed paths. The remaining branches (linkbranches), inserted individually into the network tree, define aminimum independent set of closed paths.

The matrix N is also a connection matrix. This definesthe closed paths from the independent set in the magnetic

circuit that link a closed path from the independent set inthe electric circuit and vice versa.

It is convenient for the present derivation to define a newmatrix NN such that,

N = CMNNCE (4)

where CE is the branch-mesh matrix for the electric circuit.Substituting for N in eqn. 3 from eqn. 4 and enforcing theKirchhoff's voltage and current law expressions,

ClM = 0

i =

gives

m = NNi

(5)

(6)

(7)

where / is the vector of all branch currents.The rows of both CM and CE are partitioned into tree

branch and link branch parts, with trees preceding the links.The form of the matrix NN is therefore as shown in Fig. 3.

NOTE NOLE

0

0

0

N

NOTM

NOLM

Fig. 3 Form of the matrix N^

NOTE = number of tree branches in the electric circuitNOLE = number of link branches in the electric circuitNOTM = number of tree branches in the magnetic circuitNOLM = number of link branches in the magnetic circuit

Faraday's Law is used to give the relation between the solen-oidal EMF sources e in the electric circuit and the loop fluxes07 in the magnetic circuit. This is expressed by eqn. 10 ofReference 2 which is reproduced as eqn. 8 below:

= -pNT4>j (8)

This formally expresses Faraday's law for each of theindependent sets of closed paths in the electric circuit, whenflux in the linking magnetic circuit is represented by theindependent set of closed loop fluxes 07. Substituting for Nin eqn. 8 from eqn. 4 and using the Kirchhoff voltage andcurrent law expressions:

ClE = 0

0 = C M 0 7

gives

e = -pNj,<p

(9)

(10)

(11)

where 0 is the vector of all branch fluxes.In previous work [1,2] both magnetic and electric parts of

the network model were solved for loop variables. It is equallyvalid to solve a network problem in terms of node variables[3]. The linked-network problem is therefore solved usingscalar node potentials in the magnetic circuit. The magneticscalar potential £2 is introduced by defining an alternative

IEEPROC, Vol. 130, Pt. A, No. 2, MARCH 1983 89

form of the voltage and current laws for the magnetic circuit,

M = AM£l (12)

ATM<t> = 0 0 3 )

The matrix AM is termed the branch-node matrix and itscolumns define the branches to which each node is connected.By elimination of variables, the eqns. 1, 2, 6, 7, 9 and 11—13reduce to the following set of simultaneous equations in thevariables S7 and iz:

AMPAM

CTENlPAM

ATPNNCE

d{Glp + CyxCE

+ ClNlPNNCE

ah

3 Experimental verification

The 3-dimensional eddy-current problem used to test thenode-mesh solution method was that used in Reference 2,with the conducting blocks in the moved-out position, asshown in Fig. 2C of Reference 2. A comparison of flux-densitycalculations and measurements in the z and x directions forthis problem are given in Figs. 5 and 6. Close agreementbetween calculations and measurements confirm the validityof the node-mesh method of solution of the linked-network

(14)

As a result of the zero blocks in the matrix NN, see Fig. 3,and the fact that the link-branch part of CE is a unit matrix,the product NNCE in eqn. 14 is equal to the matrix given inFig. 4.

If the matrix product NNCE in eqn. 14 is replaced by NN,where NN now has the form given in Fig. 4, then the fieldequations become:

20

AlPAM

NlPAM

AlPNN

cl(GiP + cylcE

+ Nj,PNN

Q

ii

NOLE

NOTM

NOLM

Fig. 4 Form of the product NNCE

Following a solution of eqn. 15, it is possible to obtain allcurrents in the equivalent circuit by using eqn. 6. All fluxes arefound from

Eqn. 16 is derived from eqns. 2, 7 and 12.As in the field equations for the complete mesh method

solution, (Reference 2, eqn. 12), the coefficient matrix for thenode-mesh field equations, eqn. 15 above, is complex, sym-metric and is not necessarily positive definite. The coefficientmatrix for the node-mesh field equations does, however,include the submatrix AJJPAM, the node admittance matrix,which is positive definite and can be made narrowly bandedwith consistent node ordering. This contributes significantlyto improving the condition of the field equations for the node-mesh solution from those for an equivalent all-mesh variablesolution. This is of particular importance, because in manyproblems the magnetic circuit is much larger than the electriccircuit, and thus a large part of the node-mesh coefficientmatrix is formed by the node admittance matrix.

90

Fig. 5 Z-directed flux density for the split block moved outwards

X = 70 mm, Z — 2 mmo Measured, —calculateda Magnitude, b phase

field model. Both the all mesh variable and the mixed node-mesh variable methods were found to give identical solutionsto the test problem.

4 Calculation details

The network model in the test problem was the same as thatused for the all-mesh variable solution in Reference 2. That is,the magnetic network was 6 x 6 x 3 blocks in the X, Y and Zdirections. The field equations for a mixed node-mesh solutionof this network were of dimension 130.

Calculations were performed on the UK SERC IBM 360195 computer. Solution of the field equations was performedusing the preconditioned conjugate gradient algorithm for thenormalised system as described in Reference 2. The run-timestorage requirement was 38Kbytes and the CPU time was15s.

5 Comparison of some aspects of the node-mesh andall-mesh variable methods

The condition number Xc, defined as the ratio of the maxi-mum to the minimum eigenvalue, of the coefficient matrixis a well established indicator to the ease of solution of a set

IEEPROC, Vol. 130, Pt. A, No. 2, MARCH 1983

of linear simultaneous equations. Solution is easy if the con-dition number is low. For the test problem the all meshvariable equations (see Reference 2) have a condition numberof Xc = 1420. The equivalent node-mesh equations have acondition number of Xc = 592. This represents a signifi-cantly better condition for the field equations of the node-mesh method.

The ratio r of the number of variables required to solvethe test problem by the node-mesh method to that requiredby the all mesh variable method is equal to 0.543. Thereduction in the number of variables required to solve a linked-network model, achieved by using the mixed node-meshmethod rather than the all mesh variable method, resultssolely from the use of node rather than mesh variables in themagnetic circuit. The variables used in the electric circuitare the same in both methods. Fig. 7 is presented to give someidea of the reduction in the number of variables achieved bythe use of the node-mesh rather than the all mesh method.This Figure gives the ratio of the number of nodes to the num-ber of independent meshes in a cube-shaped magnetic networkof side n, as shown in Fig. 8 for « = 2. It can be seen fromFig. 7 that a reduction of approximately 0.5—0.6 can beexpected in the number of variables for the magnetic circuitwhen the node-mesh solution is performed in preference tothe all mesh variable solution. It should be noted that thisfigure is of course problem dependent and tends to decreasewith the inclusion of equipotential planes in the mesh.

6 Comparison of number of variables required by node-mesh and differential equation field methods

It is felt that for a 3-dimensional differential method, the useof a scalar potential only in nonconducting regions is prefer-able. Only methods of this type are therefore considered. Atpresent, the majority of the differential methods for 3-dimensional problems also require the use of a scalar potentialin conducting regions, in addition to some vector quantity.For this case, both the node-mesh and the differential formu-lations require one scalar potential variable per node through-out the whole field region. A comparison, on the basis of the

number of variables, is therefore limited to a conductingregion, and the scalar potential at each node point is ignored.The conducting region chosen is cubic in shape and is sub-divided regularly into n x « x n cubes, as shown in Fig. 8for n = 2. The number of variables, NVA, required for theelectric circuit of such a region in a node-mesh solution arecompared in Fig. 9 (curve a) for several values of n, with thetotal number of vector components required for the possibledifferential formulations. NVA is found from eqn. 17 to be

NVA = ( 2 n - l ) ( / i + I)2 + 1 (17)

The first of the differential formulations, curve b, simplyassumes a method requiring one vector quantity, having threecomponents per node. Addition or deletion of variables

1.6

OJ 1 .2SZ

o 1.0

0.8

Fig. 6 X-directed flux density for the split block moved outwards - 0.6o

X — 120 mm, Z = 20 mm ,_0 Measured, — calculated j-ja Magnitude, b phase E Q.4

0.2

5 6 7n

10 11

Fig. 7 Ratio of number of nodes to number of independent meshesin a cube-shaped region of side n

Fig. 8 Regularly subdivided cube-shaped region of side n

n = 2

IEEPROC, Vol. 130, Pt. A, No. 2, MARCH 1983 91

required for the inclusion of boundary conditions has not beenconsidered in this curve. The number of variables is foundfrom eqn. 18,

NVB = (18)

Curve c assumes a differential formulation as in curve b, butalso allows for all tangential components of the vector quan-tity being zero on the surface of the conductor. An exampleof this case is the T vector [4, 5] . The number of variablesNVC is given by

NVC = 3(n + I ) 3 - 12n(n + 1) (19)

The final curve, d, assumes a differential formulation as incurve b, but also allows for all components of the vectorquantity being zero on the surface of the conductor. Anexample of this case is the R vector [6]. The number ofvariables NVD is given by

NVD = 3 ( « - l ) : (20)

5000

4000

3000

2000-

1000

10 12

Fig. 9 Comparison of number of variables required by the node-meshand typical differential formulations for a cube-shaped region of siden in the electric field

The comparison in Fig. 9 shows that the node-mesh methodis always likely to require considerably fewer variables tosolve a given subdivision than a differential method requiring,in the conductor, a scalar and vector, the components ofwhich are all nonzero on the conductor surface. If, in thedifferential formulation, two or all of the vector components

are zero on the conductor surface, the node-mesh methodwill probably require slightly more variables than the dif-ferential method, when the problem is small. For largerproblems, however, the node-mesh method is likely to requirefewer variables than the differential methods to solve a givensubdivision.

7 Conclusions

A method, using scalar node potentials in the magnetic circuitand mesh currents in the electric circuit, has been presentedfor solution of the linked-network representation of the 3-dimensional electromagnetic field. Previous solutions to the3-dimensional linked-network problem have used mesh vari-ables for both parts of the network. The new node-meshmethod was shown to require a factor of approximately 0.5—0.6 fewer variables in the magnetic network than the all meshvariable solution. In general, the node mesh equations are alsobetter conditioned than the equivalent all mesh methodequations; particularly when a large part of the field regionis nonconducting.

In comparison to the common differential equation formu-lations the node-mesh method is likely to require slightlymore variables to solve a given subdivision, when the numberof nodes in the eddy-current region is small. For problemswhere the number of nodes in the eddy-current region is large,the node-mesh method is likely to require less variables thanthe differential-equation formulations. Furthermore, thenatural symmetry of the node-mesh equations is an importantfactor in the reduction of run-time storage and, as with theconjugate gradient method, it is important in the simplif-ication of the actual method used to solve the simultaneousequations.

8 References

1 BALCHIN, M.J., and DAVIDSON, J.A.M.: 'Numerical method forcalculating magnetic-flux and eddy-current distributions in threedimensions', IEE Proc. A, 1980, 127, (1), pp. 46-53

2 DAVIDSON, J.A.M., and BALCHIN, M.J.: 'Experimental verifi-cation of network method for calculating flux and eddy-currentdistributions in three dimensions', IEE Proc. A, 1981, 128, (7),pp. 492-496

3 GUILLEMAN, E.A.: 'Introductory circuit theory' (Wiley-Toppan,1953)

4 PRESTON, T.W., and REECE, A.B.J.: 'Finite element solution of3-dimensional eddy current problems in electrical machines',Proceedings COMPUMAG Conference, Grenoble 1978, Paper 7.4

5 PRESTON, T.W., and REECE, A.B.J.: 'Solution of three dimen-sional eddy current problems: the T-n method'. IEE Colloquiumon Computer solution of three-dimensional electromagnetic fields:the state of the art, Nov. 1980, paper 2

6 BIDDLECOMBE, C.S., HEIGHWAY, E.A., SIMKIN, J., andTROWBRIDGE, C.W.: 'Methods for eddy current computationin three dimensions', IEEE Trans., 1982, MAG-18, pp. 492-497

9 Acknowledgments

The authors wish to thank the UK Science and EngineeringResearch Council for financial support and computing re-sources for the work.

92 IEE PROC, Vol. 130, Pt. A, No. 2, MARCH 1983