# Equivalent Circuits for Microwave Metamaterial Planar Components
## 1 Introduction
A classical T-model for the TLs is represented in Figure 1, comparing in (a) and (b) the configurations used to model an ordinary TL and a metamaterial structure, respectively. The change in the microwave response obtained by inverting the position of the inductor and the capacitor has been discussed in several papers and books, mainly to derive the values of the equivalent dielectric constant and the equivalent permeability of the metamaterial configuration considered as a \"material\" where the wave passes through, like in [10]. Sometimes, only inductors and capacitors are used, to simplify the approach or for educational purposes, which are enough to obtain the expected response of metamaterial structures. In this ideal case, only the so-called right-handed (RH) and left-handed (LH) TLs are considered, with consequent approximations for the definition of cut-off frequencies, the magnetic permeability values, and the dielectric constant. Nevertheless, in real cases, additional resistors and conductances must be introduced, as well as parasitic capacitances and inductors. For this reason, it is correct to have a more complete view of the problem, defining an elementary cell containing all the contributions mentioned above, leading to a composite-right-left-handed (CRLH) TL [10]. This circuit implementation is advantageous for having a balanced configuration without forbidden gaps in the dispersion relation. A careful design of propagating and resonating metamaterial devices might account for a balanced CRLH behavior because the unbalanced circuit has intrinsically a narrow-band response. From the analysis of the impedances, it turns out that, in principle, the balanced configuration is electrically matched for each frequency, leading to a broader band of operation. On the other hand, depending on the dominant response in the TL, we can always say that a branch can be mainly inductive or capacitive, using the Smith Chart's canon  
Figure 1: Transmission line equivalent circuit for (**a**) classical configuration and (**b**) metamaterial structure. L and C are the inductance and capacitance respectively. ical definitions, available, for instance, in [11; 12], and fit the equivalent circuit using experimental data. Based on this approach, simplified modeling will be discussed to determine the kind of lumped components necessary to fully describe the response of the measured TL or TL-based configuration. Similarly, filters could be studied to extract their general response as ordinary or metamaterial structures. In this paper, we shall develop a simple criterion based on analyzing the experimentally measured impedance (\(Z\)), using the definition of the \(Z\)-parameters and their application in interpreting the experimental data in actual devices. Such analysis aims to define a criterion for designing metamaterial building blocks suitable for resonant arrays and to verify their metamaterial response experimentally. Moreover, the study of configurations inspired by metamaterial geometries rarely concludes by confirming that the structure as a whole behaves like a metamaterial component or antenna by looking at the equivalent dielectric constant and magnetic permeability. For this reason, this contribution is intended to propose an experimental method to determine the RH (ordinary) or LH (metamaterial) nature of a microwave planar component focusing on the entire configuration, including implicitly parasitic lumped components and feeding networks.