Computation of the Eigenvalues for Bounded Domain Eddy-Current Models With Coupled Regions

A bounded domain of eddy-current problems leads to discrete eigenvalues and it allows one to satisfy the interface conditions between coupled regions simultaneously along two coordinates. The eigenvalues are zeros of complex domain functions that, for many eddy-current applications, have the same trigonometric form due to the sine and cosine terms in a corresponding general solution of the governing Helmholtz equation. We examine the functions' properties: 1) pole locations and multiplicity; 2) zero levelsets; and 3) region enclosing the zeros. These properties allow the construction of reliable and efficient algorithms for the eigenvalue computation, which are essential for bounded domain eddy-current models with coupled regions. We propose such an algorithm based on Cauchy's argument principle. We verify the algorithm using dimensions and materials typically encountered in diverse practical applications. The proposed algorithm performance is compared with the performance of an algorithm based on the Netwton-Raphson method.