The paper presents a homogenization method of the magnetic metamaterials, made of perpendicularly oriented resonators consisting of multi-turn planar coils. A resulting composite, in the form of parallel stripes with metamaterial cells, exhibits extraordinary properties in the medium frequency magnetic field, such as zero permeability. To identify an effective permeability of this metamaterial, two models were presented, i.e., a three-dimensional numerical model with current sheet approximation as well as Lorentz oscillator model, where individual coefficients are based on the lumped circuit parameters and directly related with a geometry of the unit cell. The accuracy of the second approach is improved by taking into account mutual inductances in a metamaterial grid. Then, a comparison is made with numerical model results to show adequacy of the adopted analytical attempt, and properties of this type of metamaterial are discussed. It is shown that discussed metamaterial structure can achieve negative permeability as well as its values, at identical resonant frequency, are dependent on number of turns of the planar coil.
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