# I-Shaped Metamaterial Using SRR for Multi-Band Wireless Communication
## 3 Theoretical Model and Equivalent Circuit of the ISMeTM  
The proposed ISMeTM metal structure was designed on an FR-4 dielectric substrate layer. The time-varying EM field changes over time. The electron density engulfs the MeTM structure. Metal bars have been used to enhance the formation of a new distinct electromagnetic wave. The sensing response is handled by the metallic ring, while the split gap between the rings provides a strong electric field for the design. The transmission line approach was utilized to analyze the ISMeTM unit cell, which allows an individual strip or patch to duplicate an RLC series circuit. The passive LC circuit, which is connected to the resonance frequency. The equivalent circuit of the proposed ISMeTM is presented in  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Dimension (mm)** & **Parameter** & **Dimension (mm)** \\ \hline sw & 10 & Rw & 1.25 \\ ls & 10 & L & 1 \\ R1 & 9 & W & 0.5 \\ R2 & 4 & g & 0.5 \\ R3 & 2.8 & sh & 1.6 \\ Rt & 0.5 & ch & 0.035 \\ Rl & 2 & tw & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter of the ISMeTM unit cell.  
Figure 1: Proposed ISMeTM: (**a**) unit cell configuration; (**b**) simulated model.  
Figure 2 which consists of the capacitance- and inductance-generating resonance on the proposed ISMeTM unit cell. The resonance frequency (\(f_{re}\)) is expressed as [33;34]:  
\[f_{re}=\frac{1}{2\pi\sqrt{L_{E}}C_{E}} \tag{1}\]  
where \(L_{E}\) and \(C_{E}\) represent the inductance and capacitance of the structure, respectively. Similarly, the split ring's inner loop's gap functions as a capacitor, while the metallic ring itself functions as an inductor. The electric resonance in the structure is created by interaction of the split ring and electric field while the interaction of magnetic fields with metallic loops during EM propagation in the structure causes magnetic resonance.  
The total capacitance between the ring gap (\(C_{g}\)) can be defined as the capacitor of the parallel plate (\(C_{pl}\)) which compensate for the fringing field by extending the parameters plate width and thickness and is expressed as [35]:  
\[C_{g}=\epsilon_{0}\frac{\textit{h}w}{g}+\epsilon_{0}(h+w+g) \tag{2}\]  
\[C_{g}=C_{fringe}+C_{pl}=\epsilon_{0}\bigg{[}\frac{(h+g)(w+g)}{g}\bigg{]} \tag{3}\]  
where \(\epsilon_{0}\) denotes the free space permittivity. \(h\), \(w\), and g represent the height, width, and gap of the ring, respectively. The inductance of the SSRR can be derived from the inductance of a rectangular loop [36]. By using a rectangular cross-sectional area with the conductor of a rectangular loop [37], we have:  
\[L=\frac{\mu_{0}}{\pi}\left\{\begin{array}{rl}(h+w)&\ln\left(\frac{2h}{R_{c} +w_{c}}\right)-h\ln\left(h+\sqrt{h^{2}+w^{2}}\right)-w\ln\left(h+\sqrt{h^{2}+ w^{2}}\right)-\frac{h+w}{2}\\ &+2\sqrt{h^{2}+w^{2}}+0.447(h_{c}\ +w_{c}\ )\end{array}\right\} \tag{4}\]  
where \(h\) and \(w\) are the height and width of the rectangular loop, respectively. \(h_{c}\) and \(w_{c}\) denote the height and the width of the conductor's cross section, respectively. The length  
Figure 2: The equivalent circuit of the proposed ISMeTM structure.  
and the width of the SSRR for the proposed ISMeTM structure \(R_{1}\) and \(R_{t}=R\), and the weighting factor \(\rho_{e}=1-g/4R\) Equation (4) is used to analyze the perimeter of the SSRR. By rewriting Equation (4) we have,  
\[L=\rho_{e}\frac{H_{0}}{\pi}\,\left\{2R\ln\left(\frac{2R^{2}}{h_{c}+w_{c}} \right)-2R\ln\left[R\left(1+\sqrt{2}\right)\right]-R+2R\sqrt{2}+0.447(h_{c}+w_ {c})\right\} \tag{5}\]  
The parameter values for \(\mu_{0}\) and \(\epsilon_{0}\) are \(8.854\times 10^{-12}F/m\) and \(4\pi\times 10^{-7}H/m\), respectively. In the equivalent circuit presented in Figure 2, C1, C2, C3, C4, C5, C6, C7, and C8 are the capacitors, and L1, L2, L3, L4, L5, L6, and L7 are the inductors. The circuit is designed using the ADS software.