Second-order formulation for the quasi-static field from a vertical electric dipole on a lossy half-space

Improved quasi-static expressions are derived for the time-harmonic electromagnetic (EM) field components excited by a vertical electric dipole (VED) lying on the surface of a flat and homogeneous lossy half-space. An analytical procedure is developed that allows to evaluate the complete integral representations for the fields, once the non-oscillating part of the integrand in the expression of the magnetic vector potential is replaced with its quadratic approximation for small values of the free-space wavenumber. The advantage of the proposed second-order quasi-static approximations resides in the possibility of relaxing the assumption of highly conducting half-space. This makes it possible to overcome the limitations implied by the previously published zeroth-order formulation, whose validity is restricted to extremely low frequencies for poorly conducting media. Numerical results are presented to illustrate the reduction of relative percent error arising from using the improved quasi-static field expressions.