## 2 Minkowski Fractal Reflectarray Design and Its Sensitivity Analysis
Hence, here, the relationship between design and performance parameters is found using the array synthesis method by exploiting the setup values of design parameters. Array synthesis is a process of designing an antenna array to achieve a desired array pattern by adjusting the individual element dimensions and their excitations. To evaluate the performance parameters for a practical scenario, the ranges of these parameters are determined. Based on these selections, the design parameters are then evaluated, and a relationship is formed among them. To find this relationship between performance and design parameters, Optimetrics from HFSS is used, where a single parameter is varied at a time while keeping all other parameters constant. After this evaluation, the values are confined to more practical and possible scenarios from a wide range of design parameters. The gain of the reflectarray usually depends on the number of elements, shape modifier, focal length, and offset feed position. The sensitivity of these design parameters is investigated with respect to the gain of the array, as shown in Figure 4. Similarly, another important performance parameter of the array, i.e., its scanning capabilities, is also exploited with respect to the most influential design parameters, as shown in Figure 5. A guideline for the design of reflectarray with a minimum to maximum range of gain and beam scanning is proposed in Table 2. The beam path of the reflectarray can be changed by adjusting the phase of the individual unit cells and the position of the unit cell in the array. The phase of the individual unit cell is a function of the fractal scaling factor and fractal iteration. Hence, ultimately, by changing the fractal scaling factor with a fixed iteration, one can be able to orient the beam path of the reflectarray. The relationship between the beam scan angle and the phase of the reflectarray is given in Equation (3). \[\varphi_{i}=k_{0}(R_{i}-sinv_{b}(x_{i}cos\varphi_{b}+y_{i}sin\varphi_{b})) \tag{3}\]  
Here, \(\varphi_{i}\) is the phase at the ith element, \(k_{0}\) is the free space wave number, \(R_{i}\) is the distance between the feed centre to the element, and \((v_{b},\varphi_{b})\) is the direction of the beam. Figure 4: _Cont._  
Figure 4: Reflectarray gain variations with respect to (**a**) Number of elements (N), (**b**) Feed/Aperture distribution efficiency (FDE/Ade), (**c**) Block Ratio (BR), and (**d**) Beam Scan Angle. A reflectarray is desired to have high gain with narrow beam radiation patterns. Hence, these two parameters are always focused while designing any reflectarray. Thus, the effect of design parameters like the number of elements (N) in the array, Feed and Aperture Distribution Efficiency (FDe/Ade), and Block Ratio (BR, Feed Diameter to Reflectarray Diameter) over the gain of the reflectarray are plotted in Figure 4. In the following Figure 5, the gain is observed with respect to changes in beam scan angles. During the analysis, all other parameters are kept constant like for variations with N (Frequency of operation: 12 GHz, RO3303 Material with 0.5 mm thickness, \(\chi\)f: 50 mm, a: 7.495 mm, BR:  
thickness, Xf: 50 mm, N: 7, a: 7.495 mm, BR: 0.2), and BR (Frequency of operation: 12 GHz, RO3303 Material with 0.5 mm thickness, Xf: 50 mm, N: 7, a: 7.495 mm, FDe/Ade: 0.6). From Figure 4, it can be observed that an increase in the number of elements of array results in an increase in the gain of the reflectarray. The normalised gain patterns for these parameter variations also confirmed the same with respect to their beamwidth changes. Similarly, the increase in FDe/ADe and BR also increases the overall gain. However, the gain variation is more in the case of changes in N rather than changes in the other three parameters. As the number of elements increases, the overall size of the reflectarray also grows, resulting in increased space requirements.