The Evaluation of an Asymptotic Solution to the Sommerfeld Radiation Problem Using an Efficient Method for the Calculation of Sommerfeld Integrals in the Spectral Domain

Abstract

A recently developed high-frequency asymptotic solution for the famous “Sommerfeld radiation problem” is revisited. The solution is based on an analysis performed in the spectral domain, through which a compact asymptotic formula describes the behavior of the EM field, which emanates from a vertical Hertzian radiating dipole, located above flat, lossy ground. The paper is divided into two parts. We first demonstrate an efficient technique for the accurate numerical calculation of the well-known Sommerfeld integrals. The results are compared against alternative calculation approaches and validated with the corresponding Norton figures for the surface wave. In the second part, we introduce the asymptotic solution and investigate its performance; we compare the solution with the accurate numerical evaluation for the received EM field and with a more basic asymptotic solution to the given problem, obtained via the application of the Stationary Phase Method. Simulations for various frequencies, distances, altitudes, and ground characteristics are illustrated and inferences for the applicability of the solution are made. Finally, special cases leading to analytical field expressions close as well as far from the interface are examined.

Through the Soil Long Range Wireless Power Transfer for Agricultural IoT Networks

“…Additionally, since the assumption of this derivation is that the soil acts as a good conductor, the electric field at the soil/air interface must be vertical due to continuity, which again assumes an electric field in the − → u θ -direction and a cos θ term equal to one. Taking (19) and dividing it by the source current (I S ) produces the equation for u e , that can be inserted into ( 5) and ( 8)…”

“…To validate (19), and with it (20), the Tx was driven at various currents (I S ) from 1A to 9A and radial measurements were taken using steel stakes as the Rx electrodes and a Tektronix THS3024 battery powered oscilloscope to quickly measure voltages between two points at the soil surface. The potential differences were measured every 2.5m, within a 50m radius around the Tx.…”

“…5(a). Using the measured Tx and Rx parameters in Table I, with a drive voltage that produced 6A in the Tx, the short-circuit Rx current was calculated from (5) The received voltage at the Rx electrodes was found to be the (19). Note that the front electrode was placed 10m from the Tx and the rear electrode was moved in 1m increments away from the Tx.…”

“…sum of the potential between the electrodes, allowing one to utilize (19) to estimate the value of received voltage based on the Rx electrode separation at any distance away from the Tx.…”

“…More obscure in present literature is the non-line-of-sight approaches where energy is transferred along the surface of the earth with a ground wave [18], [19]. Viziv Technologies is a company that claims to be working on ground wave propagation for power transfer [20], [21].…”