# A Metamaterial Inspired AMC Backed Dual Band Antenna for ISM and RFID Applications
## 2 Materials and Methods
### Unit Cell Design and Parameter Specifications  
A unit cell was designed and simulated by using CST software. The unit cell was designed to have permittivity and permeability in negative characteristic features. The designed unit cell consisted of double squared-shaped split ring resonators in which the outer ring splits on either side while the inner one was closed. Two perpendicular metal strips connected the opposite side of the closed inner ring resonator. A unit cell is generally lesser than 1/10 th of the resonating wavelength. The proposed dual band unit cell dimensional view is depicted in Figure 5, and the parametric specifications are listed in Table 2.  
Figure 6 portrays the boundary conditions of the unit cell. Such an arrangement facilitates a waveguide port, perfect electric conductor (PEC), and perfect magnetic conductor (PMC) to set perfectly on the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. The incident field propagates along the x direction with E and H fields along the y and z directions. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Dimension (mm)** \\ \hline L & 35 \\ a & 34.5 \\ b & 22 \\ c & 5.5 \\ g & 1 \\ G & 6 \\ r & 5 \\ R & 5 \\ gw & 5 \\ gl & 15 \\ gl & 0.5 \\ \(H_{t}\) & 1.6 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter Specifications of designed Unit Cell. Figure 5: Unit cell Structure, (**a**) Top view, (**b**) Rear view. Figure 6: Boundary conditions of the designed unit cell. The reflection phase results are shown in Figure 7. It can be noticed that the reflection becomes zero twice in the frequency band of interest. At low frequency, the resonance was due to the outer ring of the unit cell. At high frequency, the resonance was due to the plus-shaped structure present inside the inner ring of the unit cell. The two different null reflected phase frequencies can be independently adjusted which is highly suitable for dual band design. A wide range of metamaterial patterns have been introduced, viz., double split ring resonator, single split ring resonator (SRR) and the electric ring resonator [30]. Out of all such designs, the split ring resonator is the most used pattern due to its simple geometry. When SRR is excited with a gap perpendicular to an electric field, the resonator exhibits capacitive-inductive resonance that arises from the accumulation of charge at SRR gap area and the current at the side ring of the SRR. The suggested metamaterial had both capacitive and inductive elements. All metal bars and strips of rings acts as inductors, while the spaces between the metal bars or rings serves as capacitors. As a result of this, metamaterial unit cell created an LC resonance circuit. Figure 8 represents the equivalent circuit of the proposed unit cell structure; OC is the outer square ring and IC is the inner squared ring; L is the inductance produced in the outer squared ring and the parameters L1, L2, L3, L4 are the inductance produced in the inner squared ring of the unit cell. The parametric values of the inductor of designed unit cell structure are L = 6.2 nH, L1 = 220 uH, L2 = 1.1 nH, L3 = 2.2 nH, L4 = 4.3 nH, and the capacitance are sequentially, OC1 = 5.6 pF, IC1 = 0.51 pF, IC2 = 0.15 pF, IC3 = 0.20 pF, IC4 = 1.5 pF. Figure 8: Equivalent circuit of the unit cell. Figure 7: Reflection phase results of unit cell. The neural impedance of the FR4 substrate can be calculated [31] as  
\[Z_{sub}=\frac{\mathrm{z}_{0}}{\sqrt{\epsilon_{\mathrm{r}}}}=\frac{50}{\sqrt{4.3}} \tag{8}\]  
According to the strip line theory [32], the inductance of the strips line is calculated using  
\[L\approx 2\times 10^{-4}l\left[ln\left(\frac{l}{w+t}\right)+1.193+0.2235\left( \frac{w+t}{l}\right)\right] \tag{9}\]  
where '\(l\)' is the length of the strip line, '\(w\)' is the width and '\(t\)' is the thickness.