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FINITE ELZUENT ANALYSIS OF TRANSIENT EDDY CURRENTPBOBLEYS WITH
TIME DEPENDENT EXCITATIONSJ. Weiss V. K. GargMentor Graphics
Corporation8600 S. 1. Creekside Place Research and Development
CenterWestinghouse Electric CorporationBeaverton, OB 97005
Pittsburgh, PA 16236
ABSTRACTThis paper presents a new finite element formulation for
the solution oftransient electromagnetic probl em in
multiply-excited magnetic aystems witharbitrary time dependent
voltage and/or current excitations. T he techniquepresented in this
paper consists of obtaining a solution to Maxwellsequations in the
magnetic circuit and Kirchhoffs voltage and currentequations in the
electrical circuit given the terminal voltage and
externalimpedances. To ensure a unique definition of the terminal
voltag e th eterminal pairs is arsumed to be in a region where the
rate of change ofmagnetic flux density is negligible.The
formulation given here consists of treating the time dependent
variablessuch as the magnetic vector potent id, the electric field,
and the totalmeasurable currents as unknowns. We apply an implicit
tim e integrationtechnique to Maxwells equations, and Kirchhoffs
current and voltageequations, followed by a Galerkins projecti on
in space and obtain r tim edependent system of differential and
algebraic equations. It also isobserved that this procedure when
included in the finit e elementdiscretisation process preserves the
synmetry of the resulting coefficientmatrix. This provides a mimple
and elegant one step method for solvin g thisclass time dependent
eddy current field problems with arbitrary voltageand/or current
inputs.
Consider a magnetic circuit in series with an external
resistance E,inductance L, and excited with a time varyin g voltage
v(t), as shown inFigure 1. From Maxwells equrtions and Kirchhoffs
laws, on e writes th efollowing equations for the circuit shown in
Figure 1, [l, 21.-(l/p)V2A + U BA/& - U E, = o (1)
- U BA/& + U E, - J(t) =0 (2)Bslm + (Bext+ Lext)I(t) =V(t)
(3)
Where A is the rector potential, p is the permeability, E is the
electricfield intensity, U is the conductivity and J is the current
density. 1 isthe u ia l length of .the magnetic in the direction of
the current flm, i s
CH2731- /89/0000 - C-6 $01 OO @ 1989 IEEEI
HC-6
the unknown measurable current and v(t) is the known vol tage as
shown infigure 1.To approximate the time variation in the system of
equations (l), (2), and(3), we apply the Crank-Nicholson central
difference method, followed by aGalerkins projection in space. Then
we integrate the resulting equationsin space and obtain the
following finite element equationr :
where 9, T, Q and 0 are are real finite element coefficient
matrices and aregiven in reference [a]. The superscripts (n+l) and
n denote the (n+l)At andn(At) time levels, respectively. At
represents the time interval between twoconsecutive time instants
i. e., n and (n+l).
. I l k ) Finile Element Region1 I
Tip I-A mayneltc circuit with an ede rnal impedance and terminal
voltage V
1. Weiss, J. and Garg, V. K., Steady State Eddy Current Analysis
inMultiply-Excited Systems with Arbitrary Termina l Conditions,
IEEETrans. on Magnetics, November 1989.Garg, V. K. and Weiss, J.,
Finite Element Solution of Transient EddyCurrent Problems in
Multiply Excited Systems, IEEE Trans. onMagnetics, Hag-22,
September 1986, pp. 1267-1259.
2.
