# A New Compact Double-Negative Miniaturized Metamaterial for Wideband Operation
## 3 Methodology  
The commercially available, finite integration technique (FIT)-based CST Microwave Studio was utilized to calculate the scattering parameters. The structure is placed inside the two waveguide ports on the positive and negative \(z\)-axis. The ideal electric conductor boundary condition was characterized along the dividers perpendicular to the \(x\)-axis, the dividers perpendicular to the \(y\)-axis are characterized as perfect magnetic conductor boundaries, and a frequency domain solver was utilized for free-space simulation purposes (shown in Figure 2). Furthermore, a tetrahedral mesh with an adaptive mesh scheme was used for the proposed unit cell and array configuration investigation purposes. The standardized impedance was set to 50 ohms, and the operating frequency was set from 2 to 14 GHz.  
The reflection (\(S_{11}\)) and transmission (\(S_{21}\)) coefficients of the metamaterial unit cell and arrays were investigated to understand the electromagnetic properties of the proposed structures. The restoration of the effective medium ratio depends on the unit cell dimension, and the wave length needs to be less than the working wavelength in the media [21]. Now, the reflection coefficient (\(\Gamma\)) can be determined as follows:  
\[\Gamma=\frac{(z_{0}-1)}{(z_{0}+1)} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Dimensions (mm)** & **Parameters** & **Dimensions (mm)** \\ \hline \(a\) & 10 & \(d\) & 0.5 \\ \(b\) & 10 & \(g\) & 0.5 \\ \(l\) & 8 & \(t\) & 1.6 \\ \(w\) & 8 & \(h\) & 0.035 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design specification of the Z-shape unit cell.  
Figure 1: Metamaterial unit cell: (**a**) proposed geometry; (**b**) fabricated geometry.  
Figure 2: Simulation setup of the proposed Z-shape unit cell for CST Microwave Studio (CST MWS) in a free-space environment.  
\(z_{0}\) is the relative impedance for effective permittivity and permeability:  
\[z_{0}=\sqrt{\frac{\mu_{r}}{\varepsilon_{r}}} \tag{2}\]  
The scattering parameters \(S_{11}\) and \(S_{21}\) can be expressed as follows:  
\[S_{11}=\frac{\left(1-\varGamma^{2}\right)z}{1-\varGamma^{2}z^{2}} \tag{3}\]  
\[S_{21}=\frac{\left(1-z^{2}\right)\varGamma}{1-\varGamma^{2}z^{2}} \tag{4}\]  
Now, from \(S_{11}\) and \(S_{21}\),  
\[V_{1}=S_{21}+S_{11} \tag{5}\]  
\[V_{2}=S_{21}-S_{11} \tag{6}\]  
The Nicolson-Ross-Weir approach is utilized to separate the effective permittivity (\(\varepsilon_{r}\)) and permeability (\(\mu_{r}\)) from \(S_{21}\) and \(S_{11}\). Thus, the effective permittivity and permeability can be calculated by  
\[\varepsilon_{r}=\frac{c}{j\pi fd}\times\frac{\left(1-V_{1}\right)}{\left(1+V _{1}\right)} \tag{7}\]  
\[\mu_{r}=\frac{c}{j\pi fd}\times\frac{\left(1-V_{2}\right)}{\left(1+V_{2}\right)} \tag{8}\]  
However, the Direct Refractive strategy was used for the effective refractive index (\(\eta_{r}\)), which is calculated from the transmission coefficient (\(S_{21}\)) and reflection coefficient (\(S_{11}\)) [22]:  
\[\eta_{r}=\frac{c}{j\pi fd}\times\sqrt{\left\{\frac{\left(S_{21}-1\right)^{2}- S_{11}^{2}}{\left(S_{21}+1\right)^{2}-S_{11}^{2}}\right\}} \tag{9}\]  
For measurement purposes, a prototype of a 10 \(\times\) 10-mm\({}^{2}\) unit cell was fabricated. The measurements were performed in an open-space environment by placing the prototype inside the waveguide shown in Figure 3. An Agilent N5227A vector network analyzer was utilized to determine the scattering parameters of the modified-Z-shape unit cell. Moreover, an Agilent N4694-60001 was used for calibration purposes so that the measurements were performed accurately.