Fundamentals of Thin-Wire Integral Equations With the Finite-Gap Generator—Part I

On the Extended Thin-Wire Kernel

“…Recent studies have revealed that numerical methods applied to Hallén's and Pocklington's equations 1 with the approximate kernel yield current distributions corrupted by unphysical oscillations near the ends of the antenna and possibly near the driving point, depending on the type of the driving source [1][2][3][4][5][6][7][8][9]. These oscillations have been associated with the nonsolvability 2 of the underlying integral equations and occur when the number of basis functions becomes sufficiently larger than the length-to-diameter ratio of the wire antenna (the usual criterion of number of basis functions or points per wavelength is not relevant here).…”

“…These oscillations have been associated with the nonsolvability 2 of the underlying integral equations and occur when the number of basis functions becomes sufficiently larger than the length-to-diameter ratio of the wire antenna (the usual criterion of number of basis functions or points per wavelength is not relevant here). It is stressed that the said oscillations should not be blamed on finite computer wordlength or matrix ill-conditioning effects, which are also important but separate [1][2][3][4][5][6][7][8][9]. Matrix ill-conditioning effects occur when small perturbations (due to roundoff, errors in numerical integrations etc.)…”

“…The primary purpose of the present paper is to extend the investigations of [1][2][3][4][5][6][7][8][9] to the so-called "extended" thin-wire kernel, which was proposed by Poggio and Adams in the 1970s with the aim of improving the well-known Numerical Electromagnetics Code (NEC) [11]. The extended kernel is obtained from a certain series representation of the well-known exact kernel for thin wires by retaining its zero-order and first-order terms.…”

“…The extended kernel is obtained from a certain series representation of the well-known exact kernel for thin wires by retaining its zero-order and first-order terms. The zero-order term alone yields the approximate kernel, which has been examined in detail [1][2][3][4][5][6][7][8][9]. As pointed out in [11] for the extended kernel, "its range of validity will exceed that pertaining to the approximate kernel".…”

“…For the case of the approximate kernel, the effective-current method [4,[6][7][8][9] was proposed as a remedy to the oscillating solutions. As discussed in [12], the first paper to use the concept of an effective current seems to be [13] (its English translation is [14]).…”