# Equivalent Circuits for Microwave Metamaterial Planar Components
## 2 General Criteria to Determine the Metamaterial Nature of a Transmission Line
Despite the lack of generality because losses and parasitic contributions are not included, this criterion is accurate enough to distinguish among possible responses of the elementary cell, eventually summarizing the total DUT behavior in the case of two-port resonating structures. We must remember that an equivalent circuit is always an approximate view of a more complex structure, and sometimes, the most appropriate circuit fitting the frequency response of the DUT is not intuitive, especially at high frequencies, where distributed contributions have to be considered. In the presence of losses, the  
Figure 3: Equivalent elementary cells of an ideal TL, depending on the sign of the reflection and transmission impedances. approach maintains its generality because its imaginary part can formally substitute the above-introduced impedances, and negative or positive values of the reactive contribution can be conveniently observed using a Smith Chart representation. In formulae, for a metamaterial TL (i.e., the LH structure in the upper left corner of Figure 3) without losses, we can write the following:  
\[ \begin{array}{cc}Z_{21}=Z_{2}>0&Z_{2}=i\omega L\,\alpha\,\mu_{r}\\ Z_{1}<0\,Z_{1}=-\frac{i}{\omega C}\alpha\frac{1}{\varepsilon_{r}}\end{array}  \tag{2}\]  
So far:  
\[\mu_{r}\,\alpha\,Im\{Z_{2}\}=Im\{Z_{21}\} \tag{3}\]  
\[\varepsilon_{r}\,\alpha\frac{1}{Im\{Z_{1}\}}=\frac{1}{Im\{Z_{11}-Z_{21}\}} \tag{4}\]  
When both quantities on the right hand of the above equations, experimentally derived, are negative, we can define the TL circuit as a metamaterial structure, i.e., for a capacitive input branch and an inductive shunt branch in the T-equivalent circuit [10]. The generality of this method is based on the experimental derivation of the \(Z\)-parameters from the \(S\)-parameters, and the model can include the I/O lines bringing the signal to a filter or other resonant structures. The more general approach is based on Figure 2b, where a simple T-model is diagrammed, and the \(Z\)-circuital elements are linked to the \(Z\)-parameters obtained using the \(S\)-parameters. Of course, as discussed in the literature, the model is phenomenological and convenient because it allows the utilization of circuital concepts translated into material characteristics. Since we are not interested in a very detailed description of the circuit but in recognizing its response as an ordinary structure or a metamaterial one, any other helpful parameter to have an exact quantitative full description of the circuit can be neglected in this heuristic analysis. In conclusion, to have a TL that can be represented by a right-handed (RH) ordinary circuit or a left-handed (LH) metamaterial one in a two-port symmetric and reciprocal network is demanded for the analysis of the \(Z\)-parameters, and the sign of \(Im\{Z_{21}\}\) and \(Im\{Z_{11}-Z_{21}\}\). It might be stressed that a metamaterial structure can have additional properties, and evaluating the sign of the equivalent dielectric constant and magnetic permeability linked to the lumped components used to model the configuration is not enough to fully understand its response and novelty with respect to an \"ordinary material\". On the other hand, we are now focused on this possible characteristic of the circuit, avoiding a complete discussion implying those other non-ordinary responses, like backward radiation in the case of antennas or phase shifters.