A hybrid procedure to solve Hallen's problem

The knowledge of the electromagnetic fields in the neighborhood of an antenna needs the accurate evaluation of the current distribution on Et. This is a subject which deserves a particular attention mainly for sensors, It is called Hallen's problem the one relevant to the current distribution on a cylindrical antenna. We have already shown [1] that this problem can be formulated as a Fredholm integral equation of the second kind with a continuous kernel, and that this integral equation can be solved by a transformation into a linear system of algebraic equations. Even if this solution has a number of doubtless improvements with respect to previous approaches [2], [3], however it does not explicit the logarithmic singularity of the current due to the infinite capacitance of the infinitesimal gap, As a consequence the above expansion requires more and more terms to obtain an assigned precision of the solution, the closer we are to the gap. In this paper, we show that it is possible to extract the singular part of the current, and to obtain a reasonable precision with a finite number of terms regardless of the distance from the gap, The method seems to be suitable for thick dipole antennas, The procedure has been defined hybrid because we first resort to a finite number of steps of the iterative solution, and then the nth integral equation is solved by the Bubnov-Galerkin projection method.