# A Quad-Band Highly Selective Frequency Selective Surface with Ultra-Wideband Rejection
## 2 FSS Description and Simulation Results  
### FSS Structure Design and Equivalent Circuit Model Construction  
Figure 1 displays the proposed FSS unit structure, consisting of multiple metal layers, dielectric substrates, and air cavities, where layer 1 and layer 5 are the same winded cross-slot layers, layer 2 and layer 4 are the same square ring layers, and layer 3 is a square patch layer. The dielectric substrates of F4 BM-2 have a thickness of \(h_{1}\) and a dielectric constant \(\varepsilon\), of 2.65. The height of the air cavity was set to \(h_{2}\). Figure 2 displays the step-by-step process of design. To illustrate the forming mechanism of the quad-band, separate and joint simulations of the winded cross-slot and square ring structure were performed in curves I and II, respectively. It can be seen in curve II that two transmission poles are produced simultaneously at two metal structure resonant frequencies. Through the simulation, combination, and improvement of the structure, it can be deduced from curves I and II that two sets of resonators connected in parallel can generate four passbands. In order to obtain the suppression of the high-frequency band range, a square patch is employed in this design that is equivalent to a capacitive. To further understand the operating mechanism of the proposed FSS, the ECM was established as shown in Figure 3. The dielectric substrate with a thickness of \(h_{1}\) and an air cavity a thickness of \(h_{2}\) can act as a short transmission line with a characteristic impedance of \(Z_{d}=Z_{0}/\sqrt{\varepsilon_{r}}=231.6\)\(\Omega\) and \(Z_{0}=377\)\(\Omega\), respectively. The winded cross-slot and metallic square ring array can be equivalent to the parallel \(LC\) resonator \((L_{1}-C_{1})\) and serial \(LC\) resonator \((L_{2}-C_{2})\), respectively. The capacitor \(C_{3}\) represents the square patch. Due to the complexity of the circuit, the coupling effects between the metal layers are not  
Figure 1: (**a**) Unit cell of the proposed FSS. (**b**) Layers 1 and 5 of the unit cell. (**c**) Layers 2 and 4 of the unit cell. (**d**) Layer 3 of the unit cell. The specific sizes are \(D=6\) mm, \(l_{1}=0.55\) mm, \(l_{2}=0.57\) mm, \(l_{3}=0.625\) mm, \(w_{1}=0.15\) mm, \(w_{2}=0.57\) mm, \(d_{1}=3.86\) mm, \(d_{2}=4.85\) mm, \(h_{1}=1\) mm, and \(h_{2}=2.6\) mm. considered. By coupling among the cascaded multilayer structure, the quad-band and out-of-band suppression characteristic can be obtained. The next step is to discuss the mapping relationship between the equivalent circuit parameters and the dimensions of the FSS. First of all, the equivalent circuit parameters are derived from the curve fitting method [16]. The dimensions of the winded cross-slot layer can be optimized by the full-wave EM simulation results and the ECM results [17]. Based on [18], the equivalent circuit parameters for the square patch array can be calculated as follows:  
\[C_{3}=\varepsilon_{0}\varepsilon_{eff}\frac{2D}{\pi}\text{ln}\left(\frac{1}{ \sin\frac{\pi(D-d_{2})}{4D}}\right) \tag{1}\]  
where \(\varepsilon_{0}\) is the permeability of free space and \(\varepsilon_{eff}=(\varepsilon_{r}+1)/2\) is the relative permittivity of the dielectric substrate. The evaluations of the equivalent circuit parameters for the square ring array can be obtained using the following formula [19]:  
\[L_{2}=\frac{X_{2}}{Z_{0}\omega}=\frac{(d_{1}+2\omega_{2})F(D,2\omega_{2},\lambda )}{D\omega} \tag{2}\]  
\[C_{2}=\frac{B_{2}}{Y_{0}\omega}=\varepsilon_{eff}\frac{4(d_{1}+2\omega_{2})F(D,D-d_{1}-2\omega_{2},\lambda)}{D\omega} \tag{3}\]  
Figure 3: Established ECM of the designed FSS. \(L_{1}\) = 0.23 nH, \(L_{2}\) = 0.158 nH, \(C_{1}\) = 2.591 pF, \(C_{2}\) = 0.374 pF, \(C_{3}\) = 0.28 pF. Figure 2: Transmission coefficients in the proposed unit evolution.