# Frequency Selective Surfaces: Design, Analysis, and Applications
## 9 Miniaturized Element FSSs  
As explained in Section 5.2, in order to increase the stability of the FSS response for oblique incidences of EM wave, the unit cell size must be miniaturized [111]. This is very important whenever an FSS is used in close proximity to the radiation source. To counter this problem, a new class of FSS has been developed known as miniaturized-element FSS. In this approach, non-resonant unit cells with small dimensions are used instead of fully resonant elements in traditional FSSs. When coupled to the incident electromagnetic waves, these elements behave as lumped inductors and capacitors. The combination of inductive and capacitive elements form a resonance. The electrical size of unit cells can be decreased to less than \(\lambda/4\) or even less than \(\lambda/10\) in some cases [153, 154]. The first miniaturized-element FSS was realized by [155] as shown in Figure 11. A large array of capacitive patches is printed on one side of a thin dielectric spacer with an inductive wire grid on the other side. As the FSS is designed using miniaturized elements, its electromagnetic response can be accurately described using a lumped-element circuit model. The equivalent circuit model of this miniaturized-element FSS is given in Figure 12a, where \(L\) inductance models the wire grid in bottom layers and \(C\) represents the capacitive patch array in the top layer. The dielectric substrate is represented by a transmission line section with \(Z_{l}=Z_{0}/\sqrt{\epsilon_{r}}\), where \(Z_{0}=377\)\(\Omega\) is the free space wave impedance and \(\epsilon_{r}\) is the relative permittivity of the substrate. Based on the circuit model in Figure 12a, the top layer array produces a passband at \(f_{p1}\) described by  
\[f_{r}=\frac{1}{2\pi\sqrt{LC}}. \tag{6}\]  
Figure 12b presents the comparison of transmission coefficients obtained from EM simulations and circuit model simulations. It is evident that a good agreement for a wide range of frequencies is achieved. The scan angle performance is displayed in Figure 13 for \begin{table}
\begin{tabular}{c l c c} \hline \hline
**Article** & **Element Used** & **Size (mm)** & **Frequency (GHz)** \\ \hline \multirow{3}{*}{[141]} & Split microstrip lines & \multirow{3}{*}{10.5 \(\times\) 3} & \multirow{3}{*}{5–10} \\  & with a thin rectangular metallic bar & & \\ \hline \multirow{3}{*}{[144]} & U-shaped strip lines & \multirow{3}{*}{10 \(\times\) 10} & \multirow{3}{*}{8.4–16.2} \\  & printed on a thin substrate & & \\ \hline \multirow{3}{*}{[145]} & Circular ring unit cell & \multirow{3}{*}{34 \(\times\) 34} & \multirow{3}{*}{2.5–4.5} \\ \hline \multirow{3}{*}{[146]} & Rectangular & \multirow{3}{*}{5.4 \(\times\) 3} & \multirow{3}{*}{11–24} \\  & waveguide loaded by a wire resonator & & \\ \hline \multirow{3}{*}{[147]} & Staggered geometry of & \multirow{3}{*}{4.50 \(\times\) 1.30} & \multirow{3}{*}{11.8–13.2} \\  & metallic rectangular & & \\ \cline{1-1}  & frames & & \\ \hline \multirow{3}{*}{[148]} & Lumped capacitor & \multirow{3}{*}{7.5 \(\times\) 3.4} & \multirow{3}{*}{4.3–4.8} \\  & added with impedance & & \\ \cline{1-1}  & resonator & & \\ \hline \multirow{3}{*}{[149]} & 3D symmetric unit cell & \multirow{3}{*}{10 \(\times\) 10} & \multirow{3}{*}{12.4–16.3} \\ \cline{1-1}  & Square waveguide & & \\ \hline \multirow{3}{*}{[151]} & Slots integrated in & \multirow{3}{*}{13 \(\times\) 11} & \multirow{3}{*}{3.7–4} \\ \cline{1-1}  & square waveguide & & \\ \hline \multirow{3}{*}{[152]} & Strip lines with & \multirow{3}{*}{8 \(\times\) 8} & \multirow{3}{*}{7–8.3} \\ \cline{1-1}  & metallic plates & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of recent 3D FSS investigations. both TE and TM polarization. The structure shows a reasonably stable frequency response for angles of up to 45 degree of the incident wave. Since the electromagnetic responses of the miniaturized-element FSS can be effectively modeled using lumped element circuits, a conventional circuit-based filter theory can be utilized to design the desired filtering response.