## 2 Minkowski Fractal Reflectarray Design and Its Sensitivity Analysis
Hence, by manipulating the scaling factor of the fractal unit cell with a fixed iteration, one can vary the phase of the element, which results in the desired beam steering of the reflectarray. \[\mathcal{O}=\frac{2\pi}{\lambda}\delta \tag{1}\]  
where \(\mathcal{O}\) is the phase shift in the reflectarray, \(\lambda\) is the wavelength of the incident wave, and \(\delta\) is the patch length difference between the incident and reflected wave. The patch length difference between the incident and reflected waves can be calculated from the scaling factor using the following relationship. \[L_{n}=(1+2S)^{n}L \tag{2}\]  
Here, \(L_{n}\) is the resonating length of the unit cell generated in the nth iteration, S is the scaling factor, and L is the straight length of the unit cell. Hence, for a fixed iteration, only by varying the scaling factor of the fractal, the difference between electrical and physical length can be varied to adjust the phase of the array. The reflection characteristics of the antenna depend on the feeder of the array. The reflection characteristics of the proposed design are shown in Figure 2. In the same figure, the reflection phase characteristics with respect to the fractal scaling factor are also analysed. These curves are determined from the simulation of a unit cell using the Floquet port in CST Microwave Studio. The integral equation solver in CST simulation is used to analyse the structure. With this reflection phase and scaling factor property, the phase of each element in the array is adjusted to orient the main beam towards 30 degrees, and the radiation pattern is shown in Figure 3. From the figure, the gain of the antenna is observed to be above 20 dB with a side lobe level of less than \(-\)10 dB. From the simulation, it is observed that the gain remains relatively constant for scan angles up to 30 degrees. Above this scan angle variations, the gain will start to decrease with an increase in side lobe levels. \begin{table}
\begin{tabular}{c c c c} \hline
**Design Parameters** & **Design Values** & **Design Parameters** & **Design Values (mm)** \\ \hline N (No. of elements) & 7 & Dra (Array diameter) & 104.9 mm \\ \hline F (Focal Depth) & 83.94 mm & X-Total & 111.6 \\ \hline a (Interelement spacing) & 7.495 mm & Y-Total & 111.6 \\ \hline \(\Theta\)b (Beam direction) & 30 degrees & Z-Total & 0.5 \\ \hline Xf (Feed Distance in X-axis) & 50 mm & & \\ \hline Yf (Feed Distance in Y-Axis) & 0 mm & & \\ \hline Et (Edge Taper) & \(-\)10 dB & & \\ \hline H (Height of the substrate) & 0.5 mm & & \\ \hline \(\varepsilon\)r (Dielectric Constant) & 3 & & \\ \hline Wg (Width of the ground plane) & 111.6 mm & & \\ \hline \end{tabular}
\end{table} 
Table 1: Reflectarray optimised design parameter values with their classifications. After designing the optimised Minkowski Fractal reflectarray structure, the evaluation of performance metrics, such as gain and beam scan capabilities, is explored to the maximum extent with the help of sensitivity analysis of the design parameters in the same section. A wide variety of design parameter options are examined to find the best possible outcome of the fractal reflectarray. The simulation software, Ansys HFSS 2015.1, offers a sensitivity analysis option, which aims to identify the most influential design parameters for a specific performance parameter. It employs a second-order derivative approach in analytics to determine the graph of the relationship between the design parameter and the  
Figure 3: (**a**) Radiation pattern of the array and (**b**) in polar representation with 30-degree beam scan angle. Figure 2: (**a**) S11 vs. Frequency plot of reflectarray design, (**b**) phase vs. fractal scaling factor. performance parameter. In our case, without having prerequisites regarding the design parameters, their ranges, and the relationship between them and the performance parameters, the use of this sensitivity analysis from HFSS is not an effective approach.