A Closed-Form Solution for ELF Radiated Fields of a Line-Current Antenna in Earth-Ionosphere Waveguide

The empirical formulas for extremely low frequency (ELF) radiated fields of a line-current antenna grounding at each end in the Earth-ionosphere waveguide have been given in some literatures. But the derivation of the radiated fields in spherical waveguide is mathematically not strict. In this paper, a scalar wave function is used in mode theory of wave propagation to closely solve ELF radiated field of a vertical electric dipole (VED) located in the spherical Earth-ionosphere waveguide. The closed-form formulas for the radiated fields of a line-current antenna grounding at each end are obtained by an application of the reciprocity theorem.