# Dual-Band Single-Layered Frequency Selective Surface Filter for LTE Band with Angular Stability
## 2 Proposed Design of FSS Configuration  
### Design Methodology with Its Evolution Stages  
Figure 1 provides general design guidelines for the proposed single-band dual-frequency structure. The dimensions of the resonant patch element are thoroughly calculated using standard formulas [33; 34] for resonant frequency prediction, considering the substrate's dielectric properties. The Cruciform FSS parameters are precisely adjusted for linear polarization and dual resonance at 1.9 GHz and 2.1 GHz. The primary cross-dipole arms (L1) establish the essential framework, whilst the effective lengths of the slanted slots (L2-L5) modify the current trajectories and alter resonance frequencies due to their diminished projections. The square stubs (L6) introduce capacitive loading, which reduces resonance and improves impedance matching. Modifications in slot and stub size (W1-W4) optimize the structure's polarization and bandwidth. This meticulous design guarantees strong performance over dual frequency bands while preserving compactness and scalability. Substrate dimension calculations are as follows:  
Dielectric constant of RT/Duroid 5880 is \(E_{r}\) = 2.2  
Thickness of RT/Duroid 5880 = 0.5 mm  
Resonant frequencies are \(f_{1}\) = 1.9 GHz and \(f_{2}\) = 2.1 GHz  
Wavelength at \(f_{1}\) in the substrate will be calculated by (1). \[\lambda_{1}=\frac{c}{f_{1}\times\sqrt{E_{r}}} \tag{1}\]  
where \(c\) is the speed of light in vacuum (m/s). Approximate length of a square substrate at resonant frequency \(f_{1}\) will be calculated by  
\[L_{1}=\frac{\lambda_{1}}{2} \tag{2}\]  
The size of the substrate intrinsically depends on its resonant frequency and relative dielectric permittivity. Since the model is resonating at 1.9 GHz, its length will be dependent on it. Hence, we will consider the lower resonant frequency to calculate the resonant patch element's dimensions which gives the optimized size as \(53\times 53\) mm. To achieve a dual band, some design procedures are used which are classified into different steps to understand the operation of the proposed FSS model, which is shown in Figure 2.  
Step 1: The initial design, shown in Figure 2a, features a cross structure whose geometry can be easily modified to suit different frequencies, making it highly scalable and effective to achieve precise filtering. The configuration has a length (L1) of 45 mm and a width (W1) of 1.9 mm, providing the resonance at 2.79 GHz, as shown in Figure 3a. However, this is far from the required frequency. To achieve the desired resonance, further modifications are performed for the design to achieve the desired frequency. Step 2: The branches of the cross structure are modified, as shown in Figure 2b, to create a greater electrical length within a compact space. The configuration of a rectangular slot with dimensions of 15 mm in length (L2) and 1 mm in width (W2), which is centrally positioned rather than at the periphery of the cross structure, is performed. This position is chosen as the cross structure induces a strong surface current at the center. Consequently, this facilitates the easy flow of surface current through the structure and effectively shows the behavioral characteristic of close circuits. A constant distance is maintained between each branch to preserve the unit cell's symmetric character. Its reflection/transmission coefficient is analyzed, and the geometry provides the transmission at 2.49 GHz, as shown in Figure 3a. It does not fulfill the criteria of the proposed FSS. Hence, the model needs further modifications. Figure 1: Design methodology flowchart of the proposed dual-band single-layer structure. Step 3: In Figure 2c, the subsequent modification introduces additional branches, each inclined at an angle of 45\({}^{\circ}\) relative to each arm of the cross dipole. The inclination of the rectangular slot will effectively modify its physical length and the way it will interact with the electromagnetic field of the cross dipole.