## 2 Unit Cell Array Analysis with FEM Approach
### Validation of the Unit Cell  
The proposed unit cell is shown in Figure 4, and its dimensions are given in Table 1. The NRW method [22; 23] is chosen to verify the electromagnetic (EM) properties of the unit cell numerically. The following formulae indicate the steps to calculate and obtain the metamaterial properties by using MATLAB software.  
Figure 4: Proposed structure with dimensions.  
Figure 3: (**a**) \(S_{11}\) and (**b**) \(S_{12}\) parameter results for different array sizes of the unit cell.  
The Reflection coefficient can be written as  
\[\Gamma=\frac{\mathrm{Z}-1}{\mathrm{Z}+1} \tag{1}\]  
where \(\mathrm{Z}\) is normalized relative impedance for \(\mathrm{S}_{11}\) and \(\mathrm{S}_{12}\), and can be written as the following:  
\[\mathrm{Z}=\sqrt{\frac{\mathrm{\mu_{r}}}{\mathrm{\varepsilon_{r}}}} \tag{2}\]  
The Return loss can be represented as the following equation:  
\[\mathrm{S}_{11}=\frac{\left(1-\Gamma^{2}\right)\mathrm{Z}}{1-\Gamma^{2} \mathrm{Z}^{2}} \tag{3}\]  
The Transmission loss can be represented as the following equation:  
\[\mathrm{S}_{21}=\frac{\left(1-Z^{2}\right)\Gamma}{1-\Gamma^{2}\mathrm{Z}^{2}} \tag{4}\]  
\[\mathrm{V}_{1}=\mathrm{S}_{11}+\mathrm{S}_{21} \tag{5}\]  
\[\mathrm{V}_{2}=-\mathrm{S}_{11}+\mathrm{S}_{21} \tag{6}\]  
The relative permittivity can be represented as the following equation:  
\[\mathrm{\varepsilon_{r}}=\frac{\mathrm{c}}{\mathrm{j}\pi\mathrm{fd}}\times \left(\frac{1-\mathrm{V}_{1}}{1+\mathrm{V}_{1}}\right) \tag{7}\]  
The relative permeability can be represented as the following equation:  
\[\mathrm{\mu_{r}}=\frac{\mathrm{c}}{\mathrm{j}\pi\mathrm{fd}}\times\left(\frac {1-\mathrm{V}_{2}}{1+\mathrm{V}_{2}}\right) \tag{8}\]  
The resonant frequency of the proposed unit cell can be written as the following  
\[\mathrm{f}=\frac{1}{2\pi\sqrt{\mathrm{LC}}} \tag{9}\]  
where, the equivalent inductance in \(\mathrm{nH}\) can be calculated as the following equation:  
\[\mathrm{L}(\mathrm{nH})=2\times 10^{-4}\bigg{[}1.193+\log_{\mathrm{e}}\left( \frac{\mathrm{x}}{\mathrm{y}+\mathrm{h}}\right)+0.02235\left(\frac{\mathrm{y} +\mathrm{h}}{\mathrm{x}}\right)\bigg{]}\times\mathrm{K_{g}} \tag{10}\]  
\[\mathrm{L}(\mathrm{nH})=2\times 10^{-4}\bigg{[}1.193+\log_{\mathrm{e}}\left( \frac{\mathrm{x}}{\mathrm{y}+\mathrm{h}}\right)+0.02235\left(\frac{\mathrm{y} +\mathrm{h}}{\mathrm{x}}\right)\bigg{]}\times\mathrm{K_{g}} \tag{11}\]  
The equivalent capacitance can be calculated using the following basic capacitance equation:  
\[\mathrm{C}=\frac{\mathrm{\varepsilon_{0}\mathrm{\varepsilon_{r}}A}}{\mathrm{d}} \tag{12}\]  
Normalized relative impedance can be represented in terms of \(\mathrm{S}_{11}\) and \(\mathrm{S}_{12}\)  
\[\mathrm{Z}=\sqrt{\frac{\left(1+\mathrm{S}_{11}\right)^{2}-\mathrm{S}_{21}^{2} }{\left(1-\mathrm{S}_{11}\right)^{2}-\mathrm{S}_{21}^{2}}} \tag{13}\]  
where 'd' is the height of the structure, 'f' is the resonant frequency and 'c' is the speed of light. The copper material strips present on the unit cell can be represented as effective inductance, and the gap present between the material strips represents capacitance. The equations from 10 to 12 can be utilized to estimate the resonant frequency, where x, y and h are the length, width and thickness of the strip, respectively. Table 2 indicates that for the unit cell's S-parameters resonating frequency bands, the effective medium parameters like epsilon, mu and n have negative amplitudes. Finally, from Figure 5 and Table 2, we can conclude that the unit cell exhibits metamaterial properties for the corresponding frequencies.