# Equivalent Circuits for Microwave Metamaterial Planar Components
## 2 General Criteria to Determine the Metamaterial Nature of a Transmission Line  
A linear, symmetric, and reciprocal basic transmission line (TL) is defined in [11] and other classical books like [12]. T-sections or \(\Pi\)-sections are introduced to describe the elementary cell, using impedances to explicitly translate into inductive or capacitive contributions. The most general structure that can be plotted using this approach for a T-modeled elementary cell is shown in Figure 2.  
The structure in Figure 2b is the most straightforward configuration for a TL section. Still, it is also very general because, except for magnetic devices, most passive signal processing configurations, like resonators, filters, delay lines, or phase shifters, are electrically matched with the same input/output impedance. A microwave device's frequency response is typically achieved by measuring the so-called scattering (\(S\)) parameters, i.e., non-dimensional complex quantities computed as the ratio between input and output signals at the ports of a vector network analyzer (VNA). A VNA is an instrument that records the amplitude and phase of signals reflected from a port or passing through the device, and multi-port configurations are commercially available. The \(S\)-parameters are defined by a matrix with rows and columns whose number depends on the number of ports of the device under test (DUT). Since the DUT can be modeled in terms of impedances, an analytical approach is available to transform the \(S\)-matrix, with elements \(S_{ij}\) and \(i,j=1,2\) into the \(Z\)-matrix, with elements \(Z_{ij}\). How to pass from \(S\)- to \(Z\)-parameters is well documented in [11; 12]. Other approaches are also available based directly on the \(S\)-parameters [13]. Figure 2: General TL described: (**a**) through arbitrary impedances and (**b**) using a symmetric and reciprocal cell, with equal impedances at the input and output ports. Once we have derived the \(Z\)-parameters from post-processing the measurements, we can use the equations governing the response of the DUT in the following way:  
\[\left\{ \begin{array}{c}Z_{11}=Z_{1}+Z_{2}\\ Z_{12}=Z_{21}=Z_{2}\\ Z_{22}=Z_{1}+Z_{2}=Z_{11}\end{array} \right. \tag{1}\]  
As is well-established in microwave engineering design, the impedance analysis can be performed using the Smith Chart for an immediate visual check of the dominant contribution of the equivalent lumped component. This approach can be easily applied to the experimental data, and the qualitative result of the analysis is summarized in the following plot, where the two fundamental complex components to be considered will be \(Z_{1}=Z_{11}-Z_{21}\) and \(Z_{2}=Z_{21}\), and the imaginary part will give evidence of the main contribution in the response of the individual branches of the T-section. In practice, using a reflection and a transmission measurement for a symmetric and reciprocal TL will help to describe the two-port network's complete response. This result is immediate for a lossless section of a TL, where all the possible responses lead to the plot in Figure 3.  
The interpretation of Figure 3 is quite simple for the ideal, lossless case because there is no additional contribution to explain the measured response. It might be helpful to remember that the plot in Figure 3 is directly inspired by contributions like [1; 10], where the diagram was proposed in terms of the sign for the dielectric constant and the magnetic permeability. When \(Im(Z_{21})=Im(Z_{2})>0\) (inductive behavior), we can have \(Im(Z_{1})>0\) if an inductor is present (equivalent circuit in the upper right corner); otherwise, \(Im(Z_{1})<0\) will correspond to a capacitor (upper left corner). When \(Im(Z_{21})<0\), then \(Im(Z_{1})\) can be again negative or positive, leading to the lower left corner circuit or the lower right corner one, respectively.