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In this paper, we derive quasi-static equivalent-circuit models for the analysis and design of different types of artificial magnetic resonators-i.e., the multiple split-ring resonator, spiral resonator, and labyrinth resonator-which represent popular inclusions to synthesize artificial materials and metamaterials with anomalous values of the permeability in the microwave and millimeter-wave frequency ranges. The proposed models, derived in terms of equivalent circuits, represent an extension of the models presented in a recent publication. In particular, the extended models take into account the presence of a dielectric substrate hosting the metallic inclusions and the losses due to the finite conductivity of the conductors and the finite resistivity of the dielectrics. Exploiting these circuit models, it is possible to accurately predict not only the resonant frequency of the individual inclusions, but also their quality factor and the relative permeability of metamaterial samples made by given arrangements of such inclusions. Finally, the three models have been tested against full-wave simulations and measurements, showing a good accuracy. This result opens the door to a quick and accurate design of the artificial magnetic inclusions to fabricate real-life metamaterial samples with anomalous values of the permeability.