A singular integral representation of the electromagnetic field: Physical regularization of the ill-posedness of the near-field zones of antennas

Abstract

A classification of ill-posedness in the antenna theory is presented in the case when the basic principles of electromagnetics are not fulfilled, which is observed in the near-field radiation zones. The Tikhonov regularization of Fredholm integral equations of the first kind is compared with the physical regularization (a self-consistent method). The ill-posedness arising in the analysis of the electromagnetic field (EMF) of an electric dipole, an equidistant array on its basis, frame antennas, and a cylindrical helical antenna is considered. In particular, for the EMF of an electric dipole, the gap between the field and surface current density is removed and the limits of applicability of the method of induced electromotive forces of coupled dipoles are established. It is shown that unidirectional radiation in the azimuthal plane is typical of a ring frame antenna with a certain dimension of the ring. For a cylindrical helical antenna, the calculation of the surface current density is reduced to a hypersingular integral equation with a quadratic Cauchy singularity. The analysis of the behavior of the current flowing along the helix made it possible to explain the peculiarities of the antenna pattern for various helical turns.