On the Extended Thin-Wire Kernel

Abstract

A number of recent works have shown that moment-method solutions of Hallén's and Pocklington's equations with the approximate kernel present undesirable and unphysical oscillations near the ends of the antenna and possibly (depending on the feed model) near the driving point. Such oscillations have been associated with the nonsolvability of the underlying integral equations. They necessarily occur when the number of basis functions is sufficiently large and cannot be blamed on finite computer wordlength or matrix ill-conditioning effects. In the present paper, we show that similar phenomena occur when one uses the "extended" thin-wire kernel, originally proposed by Poggio and Adams in the 1970s and currently used by the Numerical Electromagnetics Code (NEC), which is an antenna modeling software suite that has many variants (e.g., GNEC, EZNEC Pro, SuperNeC, MiniNEC). We discuss the nature and properties of the oscillations. In particular, we show that they are milder and less rapid than those with the approximate kernel. Our main tools of study are (i) numerical investigations that carefully distinguish from effects due to roundoff, (ii) an analytical study for the case of the antenna of infinite length. We finally apply the so-called effective-current method as a remedy to the issues arising from the said oscillations. ).1 These are usually referred to as Hallén-type integral equations or thin-wire integral equations. 2 The fact that oscillations are caused by nonsolvability is conclusive for the case of the oscillations near the center because an argument that is analytical in nature has been developed in [1] and [6]. No such analytical argument exists for the case of the oscillations near the ends; here, the existing evidence is numerical (but, in our opinion, overwhelming). The present communication will re-illuminate these issues.

On the Application of Numerical Methods to Hallén's Equation: The Case of a Lossy Medium

“…We have additionally applied our methods to the so-called "extended thin-wire kernel" (see [4] for the lossless case). Our main conclusions continue to hold, while the essential benefits of the extended kernel (milder and slower oscillations) were verified via extensive numerical experiments.…”

“…For the case of the well-known approximate (also called reduced) kernel, the main difficulties arise from the fact that neither of the said integral equations has a solution; the important issue of "nonsolvability" is discussed in detail in [1]- [3]. Since 2001, the analysis of [1] has been extended in a number of directions, including the so-called "extended thin-wire kernel" [4], feeds other than the delta-function generator [5]- [8], loop antennas [9], [10], a similar equation of electrostatics [11], as well as antennas with finite conductivity, including carbon-nanotube antennas [12]. Furthermore, the analysis of [1] forms the foundation for the development of an easy-to-apply technique [11], [13]- [16] that is an a posteriori remedy for the most important difficulties.…”

On the modeling of brain fibers in the EEG forward problem via a new family of wire integral equations

“…(24) and (25), the surface potential φ is discretized with pyramid basis functions and the current density J is discretized with hat basis functions. Equations (24) and (25) are then tested with pyramid and hat functions respectively. This gives rise to the following matrix system…”

“…The reader should note that 1D formulations have been extensively studied in the context of high frequency electromagnetic modeling of wire-like structures [24][25][26], although those schemes, for perfect electrically conducting wires, are only mildly related to the ones presented here.…”

Alleviating oscillatory approximate-kernel solutions for cylindrical antennas embedded in a conducting medium: a numerical and asymptotic study

“…In this work, we further assume that |k c z 0 | ≪ 1 and |k c a| ≪ 1. Milder and less rapid oscillations occur when one employs an equally non-singular kernel, the so-called extended kernel, produced by applying a differential operator to the approximate kernel [34]. A remedy of these oscillations is addressed in detail in [11,25].…”