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.