Far-Field $kQ$ Theory for Hertzian Dipoles in Two- and Three-Dimensional Infinite Free Spaces

This study considers the electromagnetic coupling between two Hertzian dipoles located in two- and three-dimensional free spaces. First, we briefly review the generalized $kQ$ theory with respect to the reciprocal two-port impedance matrix. The $kQ$ concept is then applied to a basic 2D problem, where a pair of Hertzian dipoles is sandwiched between infinitely wide parallel conductor plates. One dipole is excited by a sinusoidal current, and the other dipole observes the voltage induced across it. Solving a 2D Helmholtz equation yields $kQ=\sqrt{\lambda/d}/\pi$, where $\lambda$ is the wavelength and $d$ is the distance between the dipoles. Similarly, we apply the $kQ$ theory to a 3D problem and obtain $kQ=3\lambda/4\pi d$. Because a Hertzian dipole is a building block for practical antennas, the results of this study will offer a theoretical foundation for developing various types of wireless power transfer systems.