Low-frequency dipolar excitation of a perfect ellipsoidal conductor

Abstract

Abstract. This paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results.

Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding

“…Indeed, present investigations [5,6] confirm that simple models as ours appear reliable when used to model the response of a general three-dimensional ellipsoid to a localized vector source in a homogeneous conductive medium both for low-contrast and high-contrast cases. However, the difficulty induced in performing analytical techniques when we are moving towards anisotropic geometrical models is strongly increasing due to the appearance of much more elaborate corresponding eigenfunctions of the introduced potentials, though the already rich literature with analytical works concerning the scattering by simple nonpenetrable metal shapes like spheres [7][8][9], spheroids [10,11], and as already mentioned ellipsoids [5,6] is open to accept new and useful analytical results. Indeed, very recently, similar analytical techniques based on differential analysis were adopted for targeting toroidal metallic objects within either a conductive surrounding, for example, Earth [12] or a lossless medium, for example, air [13].…”

“…We start from the easiest case = 3, continue to = 0, and conclude with the most cumbersome case = 2. This contribution offers a generalization of the results obtained in [5] for the particular physical application, but using the theory of ellipsoidal harmonics until a certain order ℓ = 0, 1, 2, 3 and with = 1, 2, . .…”

“…The reason for this constraint to the order was that only these few harmonic eigenfunctions were known in a closed-type analytical fashion [2]. Hence, in the aim of obtaining analytical results ready to accept further numerical implementation, the authors in [5] had to limit themselves to a particular number of ellipsoidal harmonics. Here, we provide a generalization of [5], which is the basis of a possible application of a new numerical method that could extend the range of the order up to very high values.…”

“…Hence, in the aim of obtaining analytical results ready to accept further numerical implementation, the authors in [5] had to limit themselves to a particular number of ellipsoidal harmonics. Here, we provide a generalization of [5], which is the basis of a possible application of a new numerical method that could extend the range of the order up to very high values. However, in order to apply this unique technique, we are obliged to solve the physical and mathematical problem introduced in this work from the beginning, since we wish to insert all the orders, thus the corresponding degrees, of the ellipsoidal harmonic eigenfunctions, mentioned above, that is, for ℓ = 0, 1, 2, .…”

Low‐frequency electromagnetic scattering by a metal torus in a lossless medium with magnetic dipolar illumination

“…VAFEAS methods, in order to sufficiently describe the electromagnetic process under consideration, is increasing fast instead of diminishing as one maybe expected, because effective computer algorithms must be supported by handy analytical formulae that capture the physical and mathematical nature of the scattering problem itself.To sustain such an opinion, let us refer to some collective references. The fully 3-D low-frequency electromagnetic fields, scattered from a perfectly conducting sphere in a conductive medium, excited by a known and fixed magnetic dipole, have been given in a closed analytical form, followed by a numerical demonstration in [13], while suchlike physical models with similar mathematical analysis have been introduced so as to tackle with spheroidal [14] or ellipsoidal [15] shapes, as well as with two-sphere [16] cases, the perfectly electrically conducting approximation being satisfactory in high-contrast cases, when the ratio between the body and the host conductivities is high. Motivated by the continuous interest for such kind of patterns, the author and other colleagues recently investigated the behavior of the electromagnetic fields scattered off a non-penetrable, that is, perfect conductor, ring torus, which is embedded in a homogeneous conductive medium and excited by a magnetic dipole of arbitrary orientation and harmonic time dependence, operating at the low-frequency regime [17].…”