# I-Shaped Metamaterial Using SRR for Multi-Band Wireless Communication
## 5 Methodology for Extracting the Unit Cell's Effective Medium Parameters  
The effective parameters of the proposed ISMeTM unit cell can be determined by placing the structure between two waveguides (ports) such that the electromagnetic wave (EM) has magnetic field along the \(y\)-axis and electric field in the direction of the \(x\)-axis. Thus, the wave is propagated at the \(z\)-axis. This means that the first port acts as the reflecting signal's transmitting port, while the second port acts as the signal's receiving port. In addition, a tetrahedral mesh from frequency domain solver has been used for simulating the unit cell and array structure. The impedance has been set to 50 \(\Omega\). The operating frequency range is from 2 to 18 GHz. Figure 1b shows the simulation arrangement of the proposed ISMeTM unit cell inside the waveguide. The effective medium ratio depends on the unit cell dimension and the wavelength must be less the working wavelength. The reflection coefficient (\(\Gamma\)) can be expressed as [38]:  
\[\Gamma=\frac{Z_{0}-1}{Z_{0}-1} \tag{6}\]  
Figure 4: Reflection and transmission parameters of various configurations: (**a**) S\({}_{11}\); (**b**) S\({}_{21}\), (**c**) permittivity.  
The scattering parameters \(S_{11}\) and \(S_{21}\) can be calculated as follows:  
\[S_{11}=\frac{\left(1-\Gamma^{2}\right)Z}{1-\Gamma^{2}Z^{2}} \tag{7}\]  
\[S_{21}=\frac{\left(1-Z^{2}\right)Z}{1-\Gamma^{2}Z^{2}} \tag{8}\]  
from Equations (3) and (4),  
\[V_{1}=S_{11}+S_{21} \tag{9}\]  
\[V_{2}=S_{21}-S_{11} \tag{10}\]  
where \(Z_{0}\), \(Z\), \(S_{11}\) and \(S_{21}\) represent the impedance, the interface reflection coefficient, the reflection coefficient, and the transmission coefficient, respectively. The complex electric permittivity and complex magnetic permeability can be positive or negative in different combinations [39]. The Nicolson-Ross-Weir (NRW) method is adopted to determine the effective electric permittivity (\(\varepsilon_{r}\)), magnetic permeability (\(\mu_{r}\)) and relative refractive index (\(\eta_{r}\)) of the proposed ISMeTM unit cell. The effective permittivity and permeability can be expressed as [40; 41]:  
\[\varepsilon_{r}=\frac{2}{j\pi fs_{h}}\times\left(\frac{1-V_{1}}{1+V_{1}}\right) \tag{11}\]  
\[\mu_{r}=\frac{2}{j\pi fs_{h}}\times\left(\frac{1-V_{2}}{1+V_{2}}\right) \tag{12}\]  
The refractive index \(\eta_{r}\) can be obtained as:  
\[\eta_{r}=\frac{2}{j\pi fs_{h}}\times\sqrt{\frac{\left(S_{21}-1\right)^{2}-S_{ 11}{}^{2}}{\left(S_{21}+1\right)^{2}-S_{11}{}^{2}}} \tag{13}\]  
\[\eta=\frac{1}{k_{0}s_{h}}\Big{\{}\Big{[}\Big{[}\Big{[}In\Big{(}e^{jnk_{0}n_{h }}\Big{)}\Big{]}\Big{]}+2m\pi-j\Big{[}In\Big{(}e^{jnk_{0}s_{h}}\Big{)}\Big{]} \Big{\}} \tag{14}\]  
where, \(k_{0}=\frac{w}{c}=\frac{2\pi f}{c}\) and  
\[e^{jnk_{0}s_{h}}=\frac{S_{21}}{1-S_{11}\frac{Z-1}{Z+1}} \tag{15}\]  
where \(f\), \(k_{0}\), \(m\), \(c\) and \(s_{h}\) represent the operating frequency, the wave number, the branch index, the speed of light and the substrate thickness, respectively. Equations (6)-(13) have been used to analyze the scattering parameters of the ISMeTM unit cell and array.