Input impedance of a boundary-penetrating antenna

When the analysis of an electromagnetic problem is concerned with the earth, the earth is often assumed simply as a homogeneous dissipative medium. But in the analysis of a drill-rod telemetry system, because the drill-rod is very long and buried in the earth, a more precise model of the earth is needed. Therefore, the earth should be thought as a stratified medium, at least, two-layer lossy medium, one soil and one rock. As pointed out by Long et al. (see Electronics Letters, vol.34, no.8, p.711-12, 1998), if the drill-rod itself is used as a dipole antenna excited by a /spl delta/ voltage source, the analysis of the antenna is necessary. This so-called ´boundary-penetrating antenna´ (BPA) is very special-it is so long that it stretches through the interface between the two media. The current integral equations of a BPA buried in a three-layer dissipative medium have been deduced by using the Hertz vector and the principle of superposition. In this paper, the Sommerfeld-type integrals involved in the current integral equations are simplified to closed-form formulas when the frequency used is extremely low. The physical meanings of the current integral equations are discussed. A more simple current integral equations and a numerical example are given.