Closed-form solution for underground impedance calculations

Recently, a closed-form solution for the self-impedance of a conductor directly buried in the earth has been conjectured through analogy with other closed-form approximations for overhead conductors. This letter provides a mathematical development for the equation and extends the results to the more general case of a conductor contained in a dielectric tunnel deep within a medium of arbitrary relative permeability. The derivation shows the mathematical approximations which are needed in obtaining the desired result and introduces a new simple, yet surprisingly accurate approximation for the ratio of Kelvin functions which is superior to the one-term approximation currently in use.