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Abstract
Abstract-In this paper, we present a two-step numerical technique for the investigation of magnetic fields produced by ELF sources enclosed by a ferromagnetic pipe. The first step is to employ the finite element method to solve the differential equations of the magnetic vector potential, subject to homogeneous Dirichlet boundary condition. The boundary condition is properly set on a circular surface at a distance from the outer surface of the pipe. From the solution of the magnetic vector potential, the electric and magnetic field can be computed. In particular, the fields on the outer surface of the pipe are determined. Then, in the second step, employing the equivalence principle, a computational model equivalent to the region exterior to the pipe is formulated. The equivalent currents can be determined directly from the electric and magnetic field obtained in the first step. Based on knowledge of the equivalent currents, the magnetic field outside the pipe can be computed in a straightforward manner. As an example, the two-step numerical technique is employed to investigate the magnetic field produced by an underground three-phase pipe-type cable. To validate the numerical technique, its numerical results are compared with existing measurement data. A good agreement is observed.