The propagation of waves supported by capacitively loaded loops is investigated using a circuit model in which each loop is coupled magnetically to a number of other loops. Since the coupling is due to induced voltages the waves are referred to as magnetoinductive (MI) waves. The mathematical formulations are mostly analytical thanks to long standing previous work on the magnetic and electric fields generated by currents flowing in loops. Retardation is neglected, i.e., dimensions of the structure are assumed to be small relative to the free space wavelength. The dispersion relations, derived in the most general case for a tetragonal three-dimensional structure, exhibit both forward and backward waves within a pass band. It is shown that for reproducing the salient features of the waves it is sufficient to take nearest neighbor coupling into account but coupling between loops further away must also be considered if higher accuracy is required. The investigations include that of resonances, conditions for the existence of traveling waves, tolerances, and streamlines of the Poynting vector. Waveguide components, like bends, power dividers and couplers are considered due to the potential applications of the MI waves as magnetic guides. Generality of the results, their possible implications for transverse electromagnetic wave propagation, previous work on similar waves, including the possibility of phase conjugation, are discussed in a separate section.

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The propagation of waves in a metamaterial consisting of split ring resonators ͑SRRs͒ and metallic rods is considered in several steps. The first involves the rods in isolation, the second the SRRs in isolation, and the third a combination of the two, which includes the coupling between neighboring SRRs and allows the propagation of magnetoinductive ͑MI͒ waves. The mathematical formulation is based on a conventional description of loaded transmission lines. A dispersion equation is derived to show the main features of known experimental results, including all the stop bands and passbands, the latter exhibiting both forward and backward waves. The interaction between electromagnetic and MI waves is presented in the form of a coupled dispersion equation. The applicability of the approaches based on negative material parameters is discussed.

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An equivalent circuit, consisting of bulk and distributed elements, is derived for describing the properties of a potential metamaterial element capable of providing negative effective permeability. It is the singly split double ring (SSDR), a special case of the split ring resonator (J. B. Pendry et al., IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)), obtained when the gap capacitance in the inner ring is infinitely large. The variables are the inter-ring voltage and the currents flowing in the inner and outer rings. The excitation is assumed in the form of a spatially constant temporally varying magnetic field. The functions, showing the angular variation of the variables, are found by solving a set of differential equations with boundary conditions imposed at the position of the split. It is shown from the analytical solution that the SSDR can have resonant frequencies in the full spectrum from very low to very high frequencies. It is pointed out in particular that whenever the mean diameter of the ring is equal to an odd multiple of the half wavelength it is always possible to find a set of parameters which will give rise to resonance. As examples the resonant frequencies are determined for eight sets of parameters. Results are also derived by replacing the distributed circuit with a number of discrete circuits. It is finally shown that the results obtained from the equivalent circuit model are in excellent agreement with those derived from the MICRO-STRIPES numerical package which solves Maxwell’s equations in the time domain.