IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007 2865 Equivalent-Circuit Models for the Design of Metamaterials Based on Artificial Magnetic Inclusions Filiberto Bilotti, Senior Member, IEEE, Alessandro Toscano, Member, IEEE, Lucio Vegni, Member, IEEE, Koray Aydin, Student Member, IEEE, Kamil Boratay Alici, and Ekmel Ozbay Abstract—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 in- clusions to synthesize artificial materials and metamaterials with anomalous values of the permeability in the microwave and mil- limeter-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 ex- tended models take into account the presence of a dielectric sub- strate hosting the metallic inclusions and the losses due to the finite conductivity of the conductors and the finite resistivity of the di- electrics. Exploiting these circuit models, it is possible to accurately predict not only the resonant frequency of the individual inclu- sions, but also their quality factor and the relative permeability of metamaterial samples made by given arrangements of such inclu- sions. Finally, the three models have been tested against full-wave simulations and measurements, showing a good accuracy. This re- sult opens the door to a quick and accurate design of the artifi- cial magnetic inclusions to fabricate real-life metamaterial samples with anomalous values of the permeability. Index Terms—Artificial magnetic inclusions, labyrinth res- onators, metamaterials, miniaturization, multiple split-ring resonators, split-ring resonators. I. INTRODUCTION A RTIFICIAL materials exhibiting anomalous values of the permeability, (e.g., materials with values of the relative permeability greater than one [1], with negative values of the permeability: the so-called mu-negative materials [2], with a zero value of the permeability, with an absolute value of the relative permeability less than one: the so-called mu-near-zero materials [3]) are of interest in many applications at different frequency ranges. Such materials are usually obtained at microwaves by printing suitable metallic resonating inclusions on supporting dielectric boards and stacking the boards to form a medium. Manuscript received May 2, 2007; revised July 27, 2007. This work was sup- ported by the European Union under the Network of Excellence METAMOR- PHOSE. F. Bilotti, A. Toscano, and L. Vegni are with the Department of Applied Electronics, University of “Roma Tre,” Rome 00146, Italy (e-mail: bilotti@ uniroma3.it). K. Aydin, K. B. Alici, and E. Ozbay are with the Department of Physics and the Department of Electrical and Electronics Engineering, Nanotechnology Re- search Center, Bilkent University, Bilkent 06800, Turkey. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2007.909611 The inclusions react to the impinging magnetic field, and provided that the dimensions of and the separation between the inclusions are small compared to the wavelength, the magnetic response of the artificial material can be expressed in terms of a macroscopic permeability function [4]–[6]. In most cases, the main reason to employ artificial magnetic metamaterials in the design of transmissive and radiating mi- crowave components is related to the possibility of squeezing the component dimensions [6]–[10] by using electrically small samples of artificial magnetic materials (planar slabs, cylin- drical shells, rods, etc.). Since in some cases [6]–[9] there is no theoretical limitation for the achievable miniaturization (i.e., the dimensions of the materials can be, in principle, even infinitesimal), the only intrinsic limit is represented by the dimensions of the inclusions used to fabricate the needed metamaterial samples. However, the most common type of resonator used to achieve anomalous values of the permeability at microwaves, the split-ring resonator [11], exhibits a physical dimension of the order of , which may represent, thereby, a significant limitation. In order to overcome this drawback and reduce the dimen- sions of the resonant inclusions, multiple split-ring and spiral resonators may be used. In [12], we have presented suitable circuit models for a quick design of both of these artifi- cial magnetic inclusions. Such models are limited to the ideal cases of: 1) absence of losses and 2) resonators immersed in air. Moreover, the models presented in [12] are able to accurately predict only the resonant frequency of the individual inclusions without giving any information about their quality factor, which may give a good indication of the bandwidth of operation of the metamaterial constituted by those inclusions. Finally, the circuit models proposed in [12] do not give the designer any in- formation on how to get a specific value of the real part of the permeability needed to fabricate a given component and on the losses related to the imaginary part of the permeability in real- istic layouts. The aim of this paper is to present a significant extension of the work done in [12], proposing complete circuit models that also take into account the presence of a dielectric substrate where the inclusions are printed on and the losses both in the dielectrics and metallic conductors. Moreover, in this paper, we compare the analytical results obtained from the presented models and both the full-wave simulations and measurements. On the other hand, when moving towards higher microwave frequencies, miniaturization of the inclusions might not be an 0018-9480/$25.00 © 2007 IEEE