An integral eigenvalue approach to three-dimensional field problems--A low-frequency variation of SEM

Abstract

The three-dimensional (3-D) eddy-current transient field problem is formulated first using the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u-v</tex> method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null-field integral equations and the appropriate boundary conditions germane to the problem are used to set up an identification matrix which is independent of null-field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. The eigenvalues, which can be equated with temporal response, are found to be intimately linked to the initial forcing function which triggers the transient in question. When this initial forcing function is Fourier decomposed into its respective spatial harmonics, it is possible to associate with each Fourier component a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response so obtained with the particular imposed external field governing the problem at hand. The technique is applied to the FELIX medium cylinder (a conducting cylinder placed in a collapsing external field) and compared to data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure. The technique proposed is applied in the low-frequency regime where the near-field effects must be considered. Application of the technique to a high frequency follows directly if the Coulomb gauge is adopted to represent the vector potential.