Series expansions for the electric and magnetic fields produced by a line current source

Abstract

Abstract. The electric and magnetic fields that an overhead infinitely long line current produces at the surface of the Earth can be expressed as inverse Fourier integrals over a horizontal wave number. In the quasi-static approximation acceptable in most applications and assuming a uniform Earth, expressions in terms of the Neumann and Struve functions are also possible. These are further developed in this paper, resulting in new infinite series expansions, not presented before, for the electric and magnetic fields. An application of the series expansions will remove the need to calculate the standard integral expressions, and performing exact numerical computations will be straightforward. Besides, it should be noted that calculations in the

Series expansions for the electric and magnetic fields produced by a line or sheet current source above a layered Earth

“…However, as in the case of a uniform Earth [Pirjola, 1998], numerical problems may arise when the absolute values of parameters u 1 and u 2 (equations (A5) and (A6) with k u ) are large. The fact that Sm, Cm, and gm (equations (A8)-(A10)) become very small as m increases also has to be taken into account in numerical computations.…”

“…It was shown previously that the electric and magnetic fields occurring at the Earth's surface in the fundamental model have rapidly converging series expansions [Pirjola, 1998]. …”

“…The derivation of the exact series expansions for the surface fields presented by Pirjola [1998] is based on the restrictive assumption of a uniform Earth. It is shown in this paper that by defining an equivalent propagation constant, similar expansions approximately hold true for a layered Earth, so the applicability of the series formulas is greatly extended.…”

“…In a recent paper, Pirjola [1998] derived rapidly converging series expansions for the electric and magnetic fields at the Earth's surface. The expressions are comparable to the well-known equations for the electric field presented by Carson [1926] and widely used in electrical engineering.…”

“…The series expansions are exact with the accuracy of neglecting the displacement currents. An ionospheric auroral or equatorial electrojet, which causes fluctuations of the geomagnetic field, is regarded as the main application of Pirjola [1998], as well as of this paper. The frequencies are typically in the millihertz range then, and the displacement currents do not play any role.…”