# A Polarization Independent Quasi-TEM Metamaterial Absorber for X and Ku Band Sensing Applications
## 2 Design Methodology  
An analytical model of a metamaterial absorber is a good method to describe it for better understanding of its basic design. There are different analytical models studied so far by researchers around the world. Some of them are multilayer and some are single layer. For instance, multilayer wire-based 2D thin material terminations [33] is one of the analytical method where authors characterized the design by complex surface conductivity. Full-wave numerical simulations verified this model. Metal nanoparticles have exotic optical properties and a new layered nanoparticle analytical model studied in [34] is a spherical structure for sensing applications. Another studied model is for infrared and optical regimes where the absorption was studied using both analytical and numerical methods. Here authors have used Transmission Line Theory and the Epsilon-near-zero (ENZ) property of the material to perform as an absorber [35]. This paper followed the same Transmission Line Theory to get a simplified analytical model in the GHz range. An advantage of this method is that it is a very straightforward and conventional circuit theory to develop the analysis. The proposed MM absorber consists of two ring-shaped resonators and inside those two-mirror reflexed C shape resonators are embossed on the dielectric material with a solid ground plane behind the substrate. A front view in structural representation with measurement details of the proposed absorber is depicted in Figure 1a-c. The metallic elements of the absorber are made of copper with thickness 0.035 mm, substrate thickness, \(t\) = 1.575 mm and this suggested absorber is developed on a single FR-4 printed circuit board (PCB) layer. FR-4 is a cost-effective fabrication material for application purposes. Simulation and measurement have demonstrated that the judgment, versatility and scope of work of the proposed design can be enhanced. Optimized geometrical dimensions of the proposed unit cell MM absorber is presented in Table 1.  
For microstrip MM absorber design, a transmission line model was first analyzed for a prototype shown in Figure 1a,b that is a unit cell absorber with specific dimensions. The fabricated unit cell absorber is shown in Figure 1c. Following a typical approach for microstrip circuit, inductor and capacitor specifically depend on lumped components were referred to for modeling the design. Additionally, lumped components have significant characteristics regarding their values and dependency, as they are closely related to the Q factor and resonance frequency. Moreover, as the prototype contains a ground plane, the inductance decreases as it gets closer to the ground plane [36]. In Figure 1a inductor L1 to L8 and capacitor C1 to C8 represent the equivalent microstrip circuit model. To calculate the equivalent or total inductance and capacitance, transmission line principle mentioned in [37] gives the formula:  
\[L(\mathrm{nH})=2\times 10^{-4}l[\ln(\frac{l}{w+t})+1.193+0.02235\frac{w+t}{l}]K_{g} \tag{1}\]  
\[C(\mathrm{pF})=\frac{10^{-3}\epsilon_{rd}wl}{36\pi td} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Parameter** & \(\mathbf{l_{1}}\) & \(\mathbf{l_{2}}\) & \(\mathbf{l_{3}}\) & \(\mathbf{h_{1}}\) & \(\mathbf{h_{2}}\) & \(\mathbf{h_{3}}\) & \(\mathbf{w_{1}}\) & \(\mathbf{w_{2}}\) & \(\mathbf{w_{3}}\) & \(\mathbf{w_{4}}\) & \(\mathbf{g_{1}}\) & \(\mathbf{g_{2}}\) & \(\mathbf{g_{3}}\) \\ \hline \multirow{3}{*}{**Size (mm)**} & 19 & 15 & 2.70 & 19 & 15 & 3.50 & 1 & 1 & 1 & 0.70 & 0.30 & 1.10 & 1 \\  & & & & & & Substrate length \& width, \(S\) & & & & & & \\ \cline{1-1}  & & & & & & & 20 & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimension of Ring C-shaped quasi-TEM absorber.