# Design of a Flat-Panel Metasurface Reflectarray C-Band Antenna
## 2 Operation Principle of the Reflectarray Antenna
The master-slave boundary condition and Floquet port setup were applied to build an infinite unit cell array simulation environment. The phase response depending on the size of the double ring and incident angle at 5.8 GHz is shown in Figure 3 It is clear that the relative phase shift strongly depends on the size of the ring resonator. The relative phase shift according to the incident angle is very small when the angle of incidence is less than \(+/-15^{\circ}\). The incidence/reflection angle of \(+/-15^{\circ}\) is wide enough for the LEO satellite applications because of the mission altitude of the LEO satellites. The mission altitude of the LEO satellite is typically 2000 km or less. It can cover a circular area with a diameter of 709 km. It is wide enough to cover a small country of a city. In this parametric analysis, the size of the unit cell (D\({}_{\text{cell}}\)) was fixed at 25 mm, the length of the outer ring (W\({}_{\text{ring}}\)) was varied from 10 to 20 mm and the width (T\({}_{\text{r1}}\), T\({}_{\text{r2}}\), G\({}_{\text{i}}\)) and gap between the square rings were fixed. The observed relative phase shifts in the unit cell ranged from \(-50^{\circ}\) to \(-670^{\circ}\) at 5.8 GHz. The obtained phase shifting range of \(620^{\circ}\) is wide enough to compensate the incident waves because only a phase shifting range of \(360^{\circ}\) is required to set any reflection angle. The proper dimension of the dual square ring \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{D_{\text{cell}}}\) & **25** & \(\mathbf{W_{\text{ring}}}\) & **10–20** \\ \hline W\({}_{\text{ring}2}\) & 6–16 & T\({}_{\text{r1}}\) & 1 \\ T\({}_{\text{r2}}\) & 1 & G\({}_{\text{i}}\) & 1 \\ G\({}_{\text{2}}\) & 2.76 & T\({}_{\text{s}}\) & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Unit-cell Design Parameters in mm. Figure 3: The relative phase shift of the reflected wave depending on (**a**) the size of the dual-square ring and (**b**) incident angle. Figure 2: Unit-cell geometry: (**a**) top and (**b**) side views. resonator (W\({}_{\text{ring}}\)) is 10-16 mm because it covers phase shifts of 0-360\({}^{\circ}\) at the operation frequency of 5.8 GHz. A 16 \(\times\) 16-unit cell array was designed based on the phase response of the unit cell shown in Figure 2 and Equation (4). The source horn antenna is aligned to the center of the flat panel metasurface, and the distance between the source antenna and the metasurface is 200 mm. The size of the dual square ring can be determined by calculating the required phase to compensate the incident wave on the metasurface according to (4). The size of the dual square rings and their associated phase shifting value were tabulated, and the 16 \(\times\) 16-unit cell array was placed automatically by utilizing custom-built Python code for Ansys HFSS. The design procedure is shown in Figure 4a. Figure 4: (a) Overall design procedure of the proposed reflectarray, (b) FE-BI domain decomposition simulation setup and (c) simulated radiation pattern of the designed reflectarray antenna. A simulation was run in Ansys HFSS electromagnetic simulator to verify the performance of the designed metasurface reflectarray antenna. To save computation resources and simulation time, the finite-element and the boundary integral (FE-BI) domain decomposition method was applied in this work, as shown in Figure 4b [18; 19; 20; 21]. The integration boundaries were set up around the source horn and the designed flat-panel reflectarray. The field information near the source horn antenna was integrated and applied to the reflectarray to calculate the reflected waves. The simulation result of the reflected wave is shown in Figure 4c. The designed reflection angle (\(\theta_{r}\)) of 15\({}^{\circ}\) is clearly observed.