3796 IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 10 October 1982

A ONE-STEP FINITE ELEMENT METHOD FOR MULTICONDUCTOR SKIN EFFECT PROBLEMS

J. WeissResearch and Development Center

Westinghouse Electric CorporationPittsburgh, PA

Z. J. CsendesDepartment of Electrical Engineering

McGill UniversityMontreal, Quebec, Canada

ABSTRACT The exception to computing multiple field solutionin skin effect problems is the integro-differential

A one-step procedure is developed to solve the equation approach developed by Konrad [16]. In thisskin effect problem in multiconductor busbars. The procedure, a single integro-differential equation, withprocedure differs from previous solutions in that it a single unknown (the vector potential) is employed totreats the source (or quasi-static) current density in solve the constrained multipath eddy current fieldeach conductor as an unknown. It couples the solution problem directly. While Konrad's procedure is indeedof the diffusion equation to Ampere's law, and allows elegant and is efficient compared to other procedures,for the finite element solution of skin effect problems it too has limitations: A higher level of complexity isto be obtained directly in one step from the currents inherent in the integro-differential equation approach,imposed by the power system in each conductor. and the sparsity of the finite element coefficient

matrix in the integro-differential approach is limited.INTRODUCTION

This paper presents a new formulation of theThe design of multiconductor busbars in electric steady-state multipath eddy current problem. The

devices and in power systems has been an ever-present approach taken in this paper represents a significantchallenge to the field analyst. The principal design departure from established techniques because itcharacteristic of these busbars is their dynamic treats the source current density as an unknown in theimpedance, a quantity easily obtained from a knowledge solution process. In essence, the method consists ofof the current flow paths in the busbars. However, imposing Ampere's law on the electromagnetic fieldthe solution of the current flow paths is dependent model by solving a coupled system of differential andnot only on the nature of the conductor structure, but algebraic equations for the two unknowns - thealso on the current levels imposed by the external magnetic vector potential and the source currentpower system in each separate conductor. Straight- dhnsity vector - given the total measurable currentforward field analysis of the problem gives the eddy Ik flowing in conductor C . The procedure is an easycurrent generated in each conductor from a specified k

one to understand and to implement, and provides asource current distribution; however, the combined simple and elegant one-step solution technique foreddy and source currents do not add to give the the multipath eddy current problem.correct total current values imposed by the powersystem, unless additional constraints are imposed.

THE ELECTROMAGNETIC FIELD EQUATIONSTechniques to impose power system constraints in

the steady-state multipath eddy current problem In multipath eddy current problems where chargesare varied [1-15] and include iterative procedures, and displacement currents are negligible, the steadymodal network solutions and superposition. With the state time harmonic electromagnetic field is governedsingle exception to be discussed shortly, these by Maxwell's equationsprocedures require multiple field solutions of theproblem and, therefore, cost considerably more than curl H = J (1)solutions of simple, unconstrained eddy currentproblems [15]. For example, the superposition curl E = - jwB (2)technique requires that N field solutions bedetermined for an i conductor busbar problem before V-B = 0 (3)imposing the power system constraints, the Nsolutions providing a basis for the constrained V-D = 0 (4)problem.

where E is the electric field, D is the electric flux,H is the magnetic field, B is the magnetic fluxintensity, and J is the current density vector. Leta vector potential be defined with the equation

B = curl A (5)

Equation (5) solves equation (3) exactly and may besubstituted into equation (1) to give

82 WM 102-2 A paper recommended and approved by the 1 _ _IEEE Rotating Machinery Committee of the IEEE Power curl - curl A = J (6)Engineering Society for presentation at the IEEE PES1982 Winter Meeting, New York, New York, January 31- Assuming that the vector potential and the currentFebruary 5, 1982. Manuscript submitted September 1, 1981; density vector are invariant in the longitudinalmade available for printing November 10, 1981. direction, equation (6) reduces to

0018-9510/82/1000-3796$00.75 ©D 1982 IEEE

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1 However, in this case, we know a priori that J is(div - grad A) -J (7)VP equal to a different unknown constant J in each

skThe relationship between the electric field E and the conductor C

kWe may therefore take

kto be the

vector potential A is obtained by substituting equa- k ktion (5) in equation (2) and integrating. The result function which is unity in conductor k and is zerois elsewhere

E = - ijA - Vf (8) f 1 inside conductor kak = (15)

where 4 is a scalar potential function. Let the k 0 otherwisecurrent density J be separated into two components

Applying the Galerkin process to the system of equationJ = j +Je (9) (12) one obtains

s+

e9

where J is the source current density vector and J

is the eddy current density vector related to scalar Siwq A 1 (16a)and vector potentials as follows [

J =-aV4)*l (10) ~~~~~~~~~~~~- woqt jwcxA g lI (16b)Js = aVf- 1 (10) l Uq amJ l skJ lmJ 1b

J = - jwoA (11) where A is a column vector containing the coefficients

It should be noted that neither J nor J can be Ai, S and T are matrices with elementss e

measured physically; they are introduced here for [ 3a. aa. +a. amathematical convenience. Only the total current i J x a ay ) ds (17)distribution J can be measured physically, andtherefore only J provides a meaningful description ofthe current distribution in the conductors. For astraight conductor, the impressed field E and the T= a .a.ds (18)current density J are uniform over the conductor iJ I jcross section, both in the electrostatic case and inthe quasi-stationary time harmonic case, provided the q is a column vector of Newton-Cotes quadrature weightselectric conductivity of the conductor is independentof position [17]. We may therefore combine equations q = a ds (19)(7), (9) and (11) to give 1 Ji

Am is the area of the m'th conductor, Im is the total(div - grad)A - jwaA + J = 0 (12a) current

I IJd (20)- jwA + J = J (12b) m

as the system of equations to be solved subject to the and 8sk isappropriate boundary conditions. In these equations JskA and J are the unknowns, while J is specified in the gsk- jw (21)integraf form

Equation (16b) is an expression of Ampere's lawJ ds (13)

in matrix form. It may be derived as follows: Ampere'sJ ds = I (13) law states that

k fwhere Ik is the current flowing in conductor Ck of Im Hie d (22)

cross-section Qk , k = 1, . ...N. Applying Stoke's theorem to equation (22) and sub-

The peculiar form of equations (12) is probably stituting1V for equationl(22) andnsub-the reason so many elaborate methods of solvingmultipath eddy current problems have been developed r 1before the straightforward one presented here. The 1m (div grad)A ds (23)obvious thing to do for the analyst is to substitute JAequation (12b) into equestion (12a) thus eliminating mthe unknown J from the equations; yet in this case, Finally, introducing equation (12a) into equation (23)

s ,gives equation (16b).one must not do the obvious.

Equations (16a) and (16b) are valid for a singleFINITE ELEMENT FORMULATION element in the traditional sense. To obtain the

corresponding discrete representation for the entireIn applying the finite element procedure, the problem, it is necessary to extend the assemblage of

unknowns A and J are usually approximated in terms of finite-elements in the usual way. In an N conductor-

interpolation polynominals ca. and i. as problem, thlere results

A = E A. a.i

Js z si Si (14)

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5(s + jwT) -j ra A1 r 0 1 -1 0 0 0 0 0Li jwaQt ::i [J4~~~~~'](24) -1 2 -1 0 0 0 0

l jXaQ jw J l gsS L j s = l 0 0 - 2 -l 0 0O 0 0 -1 2 -l 0

where the matrix Q has N columns, column k of whichcontains the quadrature weights q for conductor k, and O O O O -l 2 -l (28)n is an N by N diagonal matrix of conductor areas Qk. 0 0 0 0 0 -1 2The numerical formulation of the problem in thismanner results in a complex symmetric matrix where inaddition to the vector potential, unknowns g k are

introduced at each conductor or group of conductorsover which the total current is specified. 2 1 0 0 0 0 0

APPLICATION OF THE METHOD 1 4 1 0 0 0 0

The solution procedure is now illustrated by 0 1 4 1 0 0 0solving a one dimensional skin effect problem. Consider 0 0 1 4 1 0 0the infinitesimally thin slot embedded conductor shown 1in Figure 1 given in [16]. The slot is seven units T = 6 0 0 1 2 0 0 (29)long, four units of which are conductor and three 0 0 0 0 0 0 0units are air. The slot is assumed to be embedded ininfinitely permeable iron and the permeability of both 0 0 0 0 0 0 0conductor and air is equal to 1=l. The conductor sheetconductivity is o=1 and it carries a total measurablesheet current I=4 at a frequency of w=l. The boundarycondition on the vector potential is that A=O at thetop of the-air gap. 1 1

Q =L 1 1 2 ° ° (30)( AA= 0

8

By substituting the assembled matrices into (24) the7 following complex matrix equation results

X 1 Air 6 -1+i 1

I 1 3 6 0 4 0 0 12 67 05

=4 -1 2+j6 .2 0 0 o O -] |0|j2 | D

4-1_j I +O-6

2 j A5

Conductor 3 3c| A

2hisproblemare 0 0-i-i -i2 ° ° A0

o = 0 -+ 2+j-2 -(3

1~ ~ ~ ~~~ 1 2

o

0 0 0 -1 -1 0 A6 0

Fig. 1. Infinitesimally thin slot-embedded conductor. 0 0 0 0 0 -

s -: QT : The above equation yields the following results for 1he

vector potential and the source current density

_ A1 =~1 5 .18 - j 3.26q = A l l/2 l/2J (27) A2 = 15.29 - j 3.02

The assembled [S], [T], and [Q] matrices for the A = 15.39 - j 2.23seven line elements in Fig. 1 are evaluated as A4 = 14.68 - j 0.92

A = 12.0 (32)

A7 = 4.0

Js = 2.95 + j 14.74

6~~~~~~~~~~~~

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The total current density J is given by the sumof the source current density J and the eddy current | ,,Pa,

5

density -jwiA,

il 1-0.31l-j 0.44.1 0.54

2 -0.07 -j 0.56 1 = 0.56

0.72 - j90.65 = 0.97 (33)

1 2.03 - j 0.06 = 2.03

i5 2.95 +j 2.74 1 =4.03

Table 1 shows excellent agreement between thefirst order one dimensional finite element solutionof J, the second order two dimensional finite elementsolution of J and the analytical solution given inreference [16]. Consider next the three-phase busbarproblem illustrated schematically in Fig. 2. In this Fig. 3. Contours of equal magnetic vector potentialproblem, the three busbars carry currents of different magnitudes (or magnetic flux distribution)phases as set by the power system. In addition, a for the shielded three-phase busbarconducting shield surrounds the busbars on all sides, arrangement at the instant when the currentsbut carries no net current, although it does support in the conductors are I,= -500A, I IOOOA,eddy currents. Figure 3 presents a contour plot of and I = -500A.the magnitude of the magnetic vector potentialdistribution or magnetic flux distribution in thebusbars at the instant when Il= -500A, I2= lOOOA and

I= -500A. Figure 4 depicts the flux distribution El" rl -LZ> re"

32.083 msec (one-eighth of a cycle) later. At thisinstance the total current values are I = -965.9A,

I = 707.1A and I = 258.8A. These solutions have been2 3compared in some detail with analytical solutionswhere possible, and with previously publishedsolutions [13] and excellent agreement was obtained.

Fig. 4. Contours of equal magnetic vector potentialmagnitudes (or magnetic flux distribution)for the shielded three-phase busbar

I:]11L L arrangement at the instant when the currents

in the conductors are I -965.9A, I2= 707.1Al~~~~~~~~~~~~~~~~~~~~

and I3= 258.8A. 1

Analytical Solution 2nd Order 2-D Lst Order .1-D

1 -0.21 - j 0.42 -0.22 - j 0.42 -0.31 - J 0.44

2 0.005 - j 0.51 0.0035 - j 0.51 -0.07 - j 0.56

A ~~ ~~~~~~~~~~~~~30.74 - j0.55 01.73 -;4 0.56 0.72 - j0.65Fig. 2. A shielded three-phase busbar 3 .4-05 .73-05 .2-

arrangement. 1...L...I~~~~~1.96 + j 0.18 1.96 + j 0.18 .03- j 0.06

Table 1. Comparison between the new finite elementmethod and the analytical solution.

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CONCLUSION [10] M.V.K. Chari and Z. J. Csendes, "Finite elementanalysis of-the skin effect in current carrying

A new procedure has been developed to solve the conductors," IEEE Trans. Mag., Vol. Mag-13, pp.classical steady state skin effect problem. The 1125-1127, 1977.procedure is based on a coupled solution of thediffusion equation and Ampere's law. It differs from [11] C. J. Carpenter, "Comparison of alternativeprevious work because it treats the source current formulation of three-dimensional magnetic fielddensity as an unknown and allows a one step solution and eddy current problems," Proc. Inst. Elect.of the problem. It is shown that power system Eng., Vol. 124, pp. 66-74, 1977.constraints are embedded in a natural way within theelectromagnetic field model through the total [12] -, "A network approach to the numbericalmeasurable excitation current. solution of eddy current problems," IEEE Trans.

Mag., Vol. Mag-ll, pp. 1517-1522, 1975.The matrix equation generated in the procedure

is the same as that obtained with the ordinary complex [13] Z. J. Csendes and A. Konrad, "Electric machinePoisson equation solution except for the addition of modeling and power system constraints,"N rows and columns in an N conductor problem. The Presented at the Conf. on Electric Poweraccuracy and solution time of the new procedure are Problems: The Mathematical Challenge, Seattle,therefore virtually the same as that required for WA, March 18-20, 1980.ordinary finite element analysis. The solutiontechnique is illustrated by a simplified example of a [14] P. Silvester and C.R.S. Haslam, "Magnetoelluricone dimensional slot embedded conductor and a two modelling by the finite element method,"dimensional three-phase busbar arrangement. Numerical Geophys. Prospecting, Vol. xx, pp. 872-891, 1972.results for the eddy current density are in excellentagreement with those obtained by an exact solution [15] J. Weiss and Z. J. Csendes, "Finite elementand with those published in the literature. The solution of eddy currents in asynchronousmethod is simple and elegant and is valid for any generator rotors," International Conference onnumber of distinct series or parallel connected Electrical Machines, Athens, Greece, Sept. 1980.conductors of arbitrary cross sections.

[16] A. Konrad, "The numerical solution of steady-REFERENCES state skin effect problems - an integrodiffer-

ential approach," IEEE Trans. on Magnetics,[I] J. Lammeraner and M. Stafl, Eddy Currents, Vol. MAG-17, No. 1, Jan. 1981.

London: Iliffe Books, 1966.[17] P. Silvester, Modern Electromagnetic Fields,

[2] R. L. Stoll, The Analysis of Eddy Currents, Englewood Cliffs, NJ, Prentice-Hall, 1968.Oxford, England, Clarendon, 1974.

[3] C. W. Trowbridge, "Applications of integralequation methods for the numerical solutionof magnetostatic and eddy currents problems,"Proceedings of International Conference onNumerical Methods in Electrical and MagneticField Problems, Santa Margherita, Italy, 1976.

[4] J. H. McWhirter, R. J. Duffin, P. J. Brehm,and J. J. Oravec, "Computational methods forsolving static field and eddy current problemsvia Fredholm integral equations," IEEE Trans.on Magnetics, Vol. MAG-15, No. 3, May, 1979.

[5] P. Silvester, A. Konrad, J. L. Coulomb, andJ. C. Sabonnadiere, "Modal network representationof slot-embedded conductors," Elec. MachinesElectro-Mech., Vol. l, pp. 107-122, 1977.

[6] P. Silvester, "Modal network theory of skineffect in flat conductors," Proc. IEEE, Vol.54, pp. 1147-1151, 1966.

[7] - , "Skin effect in Multiple and polyphaseconductors," Presented at the IEEE WinterPower Meeting, New York, Jan. 28 - Feb. 2,1968, Paper 68TP22 PWR.

[8] - , "AC resistance and reactance of isolatedrectangular conductors," IEEE Trans. Power,App. Syst., Vol. PAS-86, pp. 770-774, 1967.

[9] 0. W. Andersen, "Finite element solution ofskin effect and eddy current problems,"Presented at the IEEE Summer Power Meeting,Mexico City, Mexico, July 1977, Paper A776616-6.

3801

Discussions does it not appear in subsequent equations?18) Why does Q in equation (26) change to C in equation (29)?A. Konrad (General Electric Company, Schenectady, NY): The paper 19) Equation (30), when solved does not yield the results given in

by the authors Dr. J. Weiss and Prof. Z. J. Csendes addresses a pro- (31). This can be seen by substituting the values of Al, A2 and Js fromblem of current interest to engineers and designers and is one of anumber of papers written on the subject by several researchers of which (31) into the first row of (30). While the matrix equation (30) followsreferences rl]-r9] are only a few examples. The theme of the present directly from equation (23), the last row and column of the coefficientLiiirences .are on y a tew examp es. ne tneme ot t 1e present matrix are incorrect.paper is a good deal similar in approach as well as in content to an in- 2 H d e ategrodifferential formulation developed by this discusser and published n20) How dd the authors obtain the 2nd order 2D solution referred toin the January 1981 and 1982 issues of IEEE Transactions on Magnetics a e< . .... . . .> ........... > ..........21) On page 4 the authors refer to an analytical solution of the[10], [11]. This discusser, however, has the following comments on dif- shielded three-phase bus bar problem shown in Figure 2. A reference toferent aspects of thne authors' paper and WIShes to seek( clarifi'cationin..ferenpt aspectsofther authors' paper and wishes to seek clarificationin the originator of such an analytical solution is in order here. A table ofrespect of others. results showing the comparison mentioned by the authors should have

1) This paper in its present form as is available at this 1982 Winter been included.Meeting suffers from a number of typographical errors, mathematical 22) The authors state that excellent agreement was obtained betweeninconsistencies and statements, in this discusser's view, not fully sup- their results for the bus bar problem and the results published in

2) In the paragraph following equation (13) the first sentence im- reference [13] of their paper. Since no flux values are given in referenceplies that theprah folowng eqauthors is the first straightfor- [13], only contour plots, how can excellent agreement be claimed? Fur-plies that the method presented by the authorsisthefirststraight thermore, the flux lines in the conductors (where skin effect occurs) areward one for skin effect problems. In view of the several paperspresented on the subject of skin effect problems and a one step single 23) in the Con sionte auo sa tha rdvariable integrodifferential method presented by this discusser in the 23) In the Conclusion the authors state that their method differsabove references [10], [11], the statement by the authors is contestable from previous work because it treats the source current density as anand at best is highly subjective. unknown and allows a one step solution of the problem.

3) On page 2, following equation (11), the authors state that the As can be seen from references [10] and [11] presented by thissource currentdensity canno .measud p y. T statement is discusser, the integrodifferential method also treats the source current

sourcecurruen density cannot be measured physically. This density as an unknown and allows a direct, one step solution of the vec-tor potential distribution. The only difference is that the vector poten-4) The plural form of the word multiconductor in the title does not t s i o f a

seem to be correct. tsal solution iSobtaied first and the source current density is computed5) The word principle should be principal in the first paragraph of separately if needed g

the Introduction. Simlarly, m the integrodifferential method described in references6) The word integrodifferential which appears several times in the [10] and [11] presented by this discusser, the power system constraints6outhewoaraphintegrouldifferenotialwhihenappears severaltimesint

are also embedded in a natural way within the electromagnetic fieldfourth paragraph should not be hyphenated. model through the total measurable excit'ation current. Moreover, the7) B is the magnetic flux density, not the magnetic field intensity as iodifferenga th o is asorali an nurodtt serie

stated after equation (4). Integrodifferential method is also valid for any 'number of distinct series8) In equation (2) E appears as a scalar quantity. or parallel connected conductors of arbitrary cross-sections as9) In equations (7) and (9) through (13) A and J appear as scalar demonstrated in reference [11] presented by this discusser.9)uIntitequations (7) and (9) through (13) A and J appear asscalar Therefore, the only difference between the authors' approach and the

10) In equation (10) z should be a subscript on the unit vector 1 integrodifferential method of [10] and [11] seems to be the idea of ob-11) In the sentence before equation (15) the word conduction should tamng the unknown source current density simultaneously with the

read conductor. vector potential distribution. The price the authros pay for this is that12) Equations (16a) and (16b) do not follow from the application of they must solve a larger matrix equation. While there may be a few12)erkin'sE rierquations(16a) and(16bdo nottefoll owfromdtheaionso more zero elements in the resulting coefficient matrix, the bandwidthGalerkin's Criterion to equations (12a) and (12b) as stated. How does

the second column of the coefficient matrix in equations (16a) and (16b) compared with the matrix obtained from the integrodifferential methodacquire the factor 9o? is larger. Unless the authors can demonstrate a definite reduction inacquire CV~~~~~~~~~~~. ~computer time, their approach offers no clear advantage over the in-

13) Without dispute, equations (12a) and (12b) are of the differential todifferetial,mth od.and algebraic type, respectively. This is in disagreement with the state- tegrodifferentialetd.ment in the last paragraph of the Introduction that a coupled system of Perhaps, et shouldby d ertionrehere that the integrodifferentialdifferential equations will be solved. On the other hand, equations (16a) method presented by thas dnscusserin references [10t and [11] is veryand (16b) appear to be the discretized equivalents of a differential equa- simply to program by anyone with a clear understanding of the aption and an integral equation, respectively. Does this mean that the proach.authors are not solving an integrodifferential system of equations?The authors seem to imply that the integrations in equation (16b) us-

ing Newton-Cotes quadrature weights are a result of the Galerkin pro- REFERENCEScess. This is incorrect. The integrations are in fact a result of the con-straint expressed by equation (13). [1] 0. W. Andersen, "Finite element solution of skin effect and eddy

14) Following the authors' line of reasoning and by using equations current problems," presented at the IEEE PES Summer Meeting,(17) through (22) one cannot arrive at the symmetric matrix equation Mexico City, Mexico, July 17-22, 1977, Paper A 77 616-6.given by (16a) and (16b). An asymmetric system is obtained. In the [2] M. V. K. Chari and Z. J. Csendes, "Finite element analysis of theform shown, both equations (16a) and (16b) as well as equation (23) are skin effect in current carrying conductors," IEEE Trans. Magn.,incorrect. vol. MAG-13, pp. 1125-1127, 1977.

15) According to the statement following equation (19) Im in equa- [3] P. Silvester, "Modal network theory of skin effect in flat conduc-tion (16b) is the total current in finite element em of area Am. Im ap- tors," Proc. IEE, vol. 54, pp. 1147-1151, 1966.pears on the right hand side of (16b). How do the authors know what [4] P. Silvester, "Skin effect in multiple and polyphase conductors,"the total current is in each finite element before actually solving the skin presented at the IEEE Winter Power Meeting, New York, Jan.effect problem? It was stated in the last paragraph of the Introduction 28-Feb. 2, 1968, Paper 68 TP 22-PWR.that the total measurable current Ik flowing in conductor Ck is the [5] P. Silvester, A. Konrad, J. L. Coulomb, and J. C. Sabonnadiere,given quantity. If the authors' answer to this is that equations (16a) and "Modal network representation of slot-embedded conductors,"(16b) are written for a single finite element (the mth one), then their Elec. Machines Electro-Mech., vol. 1, pp. 107-122, 1977.statement in the paragraph following equation (22) about assembly of [6] P. Silvester, S. K. Wong, and P. E. Burke, "Modal theory of skinthe element contributions to obtain a global representation being done effect in single and multiple turn coils," Presented at the IEEEin the usual way is incorrect.The assembly process exhibits the same Summer Power Meeting, Portland, OR, June 15-23, 1971, Paperpeculiarities as in the case of the integrodifferential approach [10], [11]. 71 TP 523-PWR.

16) With regard to the first paragraph under Application of the [7] U. Costache, "Calculation ofeddy currents and skin effect in non-Method, the infinitesimally thin slot embedded conductor shown in magnetic conductors by the finite element method," Ref. roum.Figure 1 does not appear in the book by Lammeraner and Stafi. It ap- sci. techn., Ser. electrotechn. et energ., vol. 21, no. 3, pp. 357-363,pears in reference [10] presented by this discusser. 1976.

17) What is the significance of the symbol A in equation (26)? Why [8] N. Mullineux, J. R. Reed, and I. J. Whyte, "Current distribution

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in sheet- and foil-wound transformers," Proc. Inst. Elec. Eng., In comparing this paper with that of Konrad (16), this discusser findsvol. 116, no. 1, pp. 127-129, Jan. 1969. no significant differences in technical approach, only differences in ex-

[9] J. Donea, S. Giuliani, and A. Philippe, "Finite elements in the ecution. Konrad's approach yields a fuller symmetric matrix, while thissolution of electromagnetic induction problems," Int. J. Numer. approach yeilds a matrix with a banded structure reminiscent of finiteMeth. Eng., vol. 8, no. 2, pp. 359-367, 1974. difference potential problems. The usual banded matrix solver algo-

[10] A. Konrad, "The numerical solution of steady-state skin effect rithms will perform equally well (poorly?) with either approach,problems - An integrodifferential approach," IEEE Trans. although some of the "skyline" solver algorithms might prefer the ap-Magn., vol. MAG-17, pp. 1148-1152, 1981. proach given in this paper. Newer conjugate gradient method solvers

[11] A. Konrad, "Integrodifferential finite element formulation of would be equally happy with either approach. The choice of which onetwo-dimensional steady-state skin effect problems," IEEE Trans. to use is dependent upon the user's computer resources.Magn., vol. MAG-18, pp. 284-292.

Manuscript received February 11, 1982. REFERENCES

[A] A. Konrad, "Integrodifferential Finite Element Formulation ofTwo-Dimensional Steady State Skin Effect Problems", IEEETrans. on Magnetics, VOL. MAG-ll, No. 1, January 1982, pp.

J. A. Mallick (General Electric Company, Schenectady, NY): The skin 284-292.effect problem in current carrying conductors illustrates the difficulty incoupling a two dimensional field solution with an external circuit con- Manuscript received February 18, 1982.straint. Too often one blindly solves field problems while forgetting theoriginal assumptions implicit in the problem statement. Since the "in-finite length" two dimensional assumption can be replaced by a finitelength problem bounded by two infinitely conducting planes, the im- .Weiss and Z. J. Csendes: The authors are pleased by the interest ex-plicit assumption here is that all conducting members are shorted at the pressed Zt Jcsensin outhorsartunate, hever, that"end",werevr thse my b. Th us.fcntan qain,sc pressed by the discussions in our work. It is unfortunate, however, that'sends", wh1erever th1ese may be. The use of constraint equations, suchnasis done here,allowsonetomodfythese"connectio mathemati- they were not able to read the paper in the proper perspective and,

cally at the expensl of increased prodblet complexity this is the" ri f hence, did not understand the significance of the new approach.carryiat thecuite neofyincrfiead perobl. cmextthis istheped As each of us has stated, the multiconductor skin effect problem hasmarrying circuit theory and fild thzeory. whnere these self-consistent ed- been solved many times in the literature. The significance of the new ap-dy current solution methods will pay off is in the solution of magneti- prac s tati tres theq ity (alledThe source urrentdnically nonlinear problems, and this discusser looks forward to seeing fur- roach IS that It treats the euantit tJs called the source current densit

the.r work in this area. in the paper) for the first time as an explicit unknown, on the same levelthis pro of skin

a efi as the vector potential A, and not as a second quantity. While one canThis problem of skin effects in current carryAngconductorsef.s(16 argue that Js appears implicitly in other previous formulations, we

addressed)ndthis discusserasrwellommeindsthatw papers byA.K drefd i6 maintain that treating Js as an explicit unknown on par with the vectorand (A)), and this discusser recommends that these papers by read in potential A, as is done in this paper, has the triple advantage of simpli-parallel so that the reader can appreciate the differences in the ap- fying the problem formulation, of reducing the complexity of the com-proach. This present paper unfortunately suffers from a rather large puter code, and of speeding up the computation.number of basic technical and presentation errors. To put the present work in the proper perspective, we must considerOne should note that the equation system given by equation (16) is an as the background to both this paper and to the integrodifferential

incorrect representation of the differential equation and constraint technique, the superposition solution technique published by Csendesequation given in (12), as a simple matrix multiplication will quickly and Konrad [13]. In the superposition approach, the skin effect pro-show. The right hand column of the system matrix contains the ex- blemtraneous multiplier jcot; its removal leaves an equation system which isconsistent. Unfortunately this error is propagated throughout the rest div - grad A - jwaA = -J (dl)of the paper, yielding a matrix equation (30) which is not satisfied by (dsthe solution vector given (31). is solved by creating a set of basis solutions [Ak], where each AkWhen the extraneous jcog term is removed, the matrix equation (16) is satisfies the equation

made nonsymmetric, but it can be made symmetric by rewriting equa-tion (12b) as: -1 inside conductor k

(D2)-A + - 0 otherwise.

jwa JX Imposing the constraintThis discusser feels that the term "Galerkin process" is misleading J - - (D3)

when applied to the resulting equation system (16). Inspection reveals integrated over each conductor cross-section then provides a matrixthat only equation (12a) is subjected to the Galerkin method usually ap- equation for the multipliers ck in the superposition solutionplied in finite element approximations; that is, the residual equation ismultiplied by the finite element approximation polynomials and inte- Ngrated over the solution space. Equation (16b) is obtained from (12b) by A = cAa simple integration over the element area, which is not a true Galerkin k=1 (D4)operation; the term "collocation" might be more appropriate here.Equation (22) is missing a minus sign. Equation (23) has the ex- It is shown in reference [13] that the coefficients ck are identical to the

traneous jCo" multiplier propagated from equation (16). On page 3, row source current densities Jk in each conductormatrix "C" appears in equation (29), but this discusser cannot seewhere this fits into the asssembly of the matrix equations. As mentioned ck = Jk ,k = 1, .... N (D5)previously, matrix equation (30) is not satisfied by the values given inequation (31); however, if one divides the last column and last row of It may therefore be argued that reference [13] is the first paper in whichthe system matrix by "j", one can obtain the solutions given in (31). Js is treated as an unknown, although the superposition technique is

Table 1 on page 4 mentions a second order, two dimensional solution clearly a circuitous route for computing it.which does not appear in the paper; I can only assume that it must be The next development in the solution of the skin-effect problem isfor the same slot embedded conductor. Perhaps the authors can Dr. Konrad's integrodifferential technique in which equations (Dl) andelaborate. (D3) are solved by eliminating Js from the equations and replacingThe three phase bus bar solutions given in figures 3 and 4 would be equations (Dl) and (D3) by a single integrodifferential equation. While

more useful if some dimensions and conductivities were given. As it is, Js may be computed from the integrodifferential solution after com-the skin effect in the bars is difficult to evaluate. The bars appear to be puting A as Dr. Konrad states, the fact remains that the integrodifferen-excluding all the flux, and might just as well have been represented by tial approach does not solve for A and Js from a coupled system ofsome surface currents. equations.

3803

We may summarize the three competing procedures for solving skin Teffect problems as follows: 01

DIRECT NUMERICAL SOLUTION OF

A i~s - and integrate. This givesSuperposition no yesIntegrodifferential yes noPresent paper yes yes

In hindsignt, of course, the approach presented in this paper, that of ;~2asio & d a ~~solving equations (Dl) and (D3) directly as a coupled system of equa-tions for A and Js, appears to be so obviously the logical choice onewonders why it wasn't done first. However, it is not proper to claimafter its publication that there is "no significant difference in technicalapproach" between the simple direct solution and the complicated in- _jwo T s wdirect one - and then to pretend that the complicated procedure includ- fds jw dsed the simple one as well. If the present technique represents the Solu- LJ Ltion of an "integrodifferential system of equations" because it solvesequations (DI) and (D3) then so does the superposition technique whichalso solves equations (Dl) and (D3) and also uses quadrature formulasto evaluate the integrals. By this fallacious reasoning one would con--clude that the integrodifferential technique is included in the superposi-

0

tion method, since the superposition approach was developed first. One0gets the feeling of "sour grapes" from Dr. Konrad's discussion: He canI

authors pay. .".The fact is that he did not think of it.The remaining substantial criticism by the discussors is their confu-

sion with regard to Galerkin's method. Since Galerkin's method is so Introducing equations (15) and (13) into (D8) gives equation (16).well known, we had omitted some of the details in the derivation. The point to emphasize in the above is that it is a "straight" GalerkinHowever, since there appears to be some doubt in the application of procedure. The discussors are wrong in thinking that the GalerkinGalerkin's method in this case, we are happy to fill in the details. technique may be applied without integrating over the solution region.

In matrix form, equations (12) are: A Galerkin method for modeling a differential or algebraic equationwithout integrating over the problem region is a contradiction in terms.

r -~~~ As can be seen above, the integral f1gk Jds is inherent in the GalerkinV2- jwo -jwoy A I process and is not generated ad hoc. Indeed, one of the beauties of the~~~~~~presentapproach is the natural way in which the constraint equation(13) is embedded in the analysis.

L~~~WO i L (D6)~~~~~~~~~~~ It is unfortunate that some minor typographical errors crept into the

I.jW jwo- LI jJ 6 Winter Meeting manuscript. These have been corrected. We regret thatthe letter J in the place of g in equation (16) made it appear that thisequation was not symmetric and, that the numbers in equation (31) did

Corresponding to equation (14), the unknowns are approximated as not add up. (Equation (16) is indeed symmetric and the numbers do in-deed add up.)The second-order, two-dimensional finite element solution of the

rA 0 ~ A] slot-embedded conductor was obtained by running a second-order,

1 II ~~~~~~~~~two-dimensional finite element computer program with NeumannI = II ~~~~~~~~~~~~boundaryconditions on the sides. We did not say that an analytical

(D) solution of the shielded three-phase bus bar problem exists. The com-

slot-embedded conductor problem. We are puzzled by Dr. Konrad'squestion concerning the comparison of flux values with the solutiongiven in reference (13]. Doesn't he remenber that one of the authors

where a] and pare row vectors containing the approximation functions (Csendes) was his coauthor on this paper, and his friend?al and P~i of the paper. In standard Galerkin form, we substitute (D7)into (D6), pre-multiply by: Manuscript received April 15, 1982.