Effective magnetic permeability measurement in composite resonator structures

This paper studies the measurement of magnetism in the structured arrays of resonators. In a recent study, we showed how the unit-cell setup is a representative of the magnetic behavior of a whole structure composed of sufficiently long resonators. Here, we investigate the limits and the validity of such a setup for the characterization of metamaterials composed of finitely long resonators having a length that is smaller than the cell dimensions. According to Lorentz's theory, we show the spatial regions over which to average the effective field, and we derive the expression of the effective magnetic permeability for the unit-cell setup and for an extended structure. To test our conjecture, we compare experimental results that are performed on the unit-cell setup to solutions of an equivalent circuital model and to measurements that are directly obtained on a 3 × 3 structured array. Finally, we comment on the operative meaning of the measurement of the effective magnetic permeability in composite media.