General expressions and physical origin of the coupling coefficient of arbitrary tuned coupled electromagnetic resonators

The theory and operation of various devices and systems, such as wireless power transfer via magnetic resonant coupling, magneto-inductive wave devices, magnetic resonance spectroscopy probes, and metamaterials can rely on coupled tuned resonators. The coupling strength is usually expressed in terms of the coupling coefficient kappa, which can have electrical kappa(E) and/or magnetic kappa(M) components. In the current article, general expressions of kappa are derived. The relation between the complex Poynting equation in its microscopic form and kappa is made and discussed in detail. It is shown that kappa can be expressed in terms of the interaction energy between the resonators' modes. It thus provides a general form that combines the magnetic and electric components of kappa. The expressions make it possible to estimate the frequencies and fields of the coupled modes for arbitrarily oriented and spaced resonators. Thus, enabling the calculation of system specific parameters such as the transfer efficiency of wireless power transfer systems, resonator efficiency for electron spin resonance probes, and dispersion relations of magneto-inductive and stereo-metamaterials structures. (C) 2015 AIP Publishing LLC.