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

In a recent study, we proposed improved quasi-static expressions for the electromagnetic field components excited by a vertical electric dipole lying on the surface of a flat and homogeneous lossy half-space. The present paper introduces an analytical approach to derive analogous formulas for the case of the horizontal electric dipole. The procedure is based on the expansion of the integral representations for the fields into power series of the ratio between the wavenumbers in free-space and in the conducting medium. Later, the terms in the expansions up to the second order can be reduced to known tabulated integrals. Numerical results are presented to illustrate the improvement in accuracy that follows from using the second-order approximations for the fields in place of the zeroth-order ones. In the quasi-static frequency range and beyond, use of the new formulation makes it possible to reduce the maximum relative error in the calculation of the fields from about 23% down to less than 7%.