## 3 Results and Discussion
### Design of Unit Cell and Cascade Connection Analysis of Corrugated PPW Structures
This limitation significantly reduces the computational burden in achieving the design requirements. In the design space, the stopband edges of the first stopband region for each parameter are found and recorded using \(X_{+}\) and \(X_{-}\) functions. We obtained a significant dataset on a large scale, determining the band edge frequencies determined by the AFGSM functions in the constrained design space. The dataset contains a variety of selectable stopband frequencies and the stopband widths situated at these centers. We have selected a center frequency in this context that can cover the targeted stopband. From this set of data, Equation (5) pulls out the values for different  
Figure 10: (**a**) Designed bandgap via AFGSM functions. (**b**) Amplitude–frequency characterization of cascading different numbers of the designed single C-PPW unit cell via CST, HFSS, and EC results. dimensions with a center frequency of 10 GHz, different stop bandwidth values, and a certain frequency deviation (\(\delta\)), as shown in Table 2. The AFGSM method offers various design dimensions for identifying appropriate unit cell parameters via multi-geometric parameter optimization, as illustrated in Table 2. The crucial aspect prior to dimensional design, as illustrated in the dataset of the second example, is to restrict the design area to a defined region. In this case, the AFGSM approach guarantees the quick and precise operation of the design algorithm. At this point, three scenarios, bold and italicized in Table 2, have been selected to meet the design requirements of the dimensional design applicability of Equation (5) for the same and different stop bandwidths. Figures 11-13 illustrate dispersion diagrams and the dominant mode of the \(|S_{21}|\) frequency characteristics derived from various finite periodic configurations of designed unit cells throughout three selected scenarios. All AFGSM function behaviors and dispersion characteristics of all scenarios given in Figures 11a, 12a, and 13a indicate that the roots of the AFGSM functions can efficiently seperate passband/stopband regions. The analysis of the filtering characteristics obtained by cascading the unit cells was performed in CST Microwave Studio and HFSS simulation environments. In order to model one-dimensional corrugated PPWs in CST and HFSS, it was necessary to assign appropriate boundary conditions. For this purpose, in CST, electric (\(E_{t}=0\)) and magnetic (\(H_{t}=0\)) conditions were used; in HFSS, perfect E and perfect H boundary conditions were assigned to the top, bottom, and side walls of the considered structures, respectively. The same computer was used to run the simulations of CST, HFSS, and our code of the proposed method. The ordinary laptop had an Intel (R) Core (TM) i7-6700 HQ [EMAIL_ADDRESS] GHz with 16 GB RAM. In the examined structures, the first ten dominant waveguide modes of excitation were satisfied to perform with the waveguide port in CST and with the waveport in HFSS,  
\begin{table}
\begin{tabular}{l l l l l} \hline \(\Delta f\) & \(\delta\) & \(a\) & \(h_{2}=h_{3}\) & \(b\) \\
**[GHz]** & **[MHz]** & **[mm]** & **[mm]** & **[mm]** \\ \hline
1.8 & 30  
and the frequency characteristics of the first dominant mode of \(|S_{21}|\) were investigated. Obtaining the dataset in the given limited design space took 360 h with CST Eigen Mode Solver, while solving the eigenvalue equation based on the circuit model took 612 s. The same dataset was generated in 20 s using the AFGSM method. We observe from Figure 11a that Scenario 1's band edges were 8.648 GHz and 11.344 GHz. In the first scenario, we selected a stop bandwidth of 2.7 GHz and determined the dimensions using the AFGSM functions, with a deviation of 10 MHz. The filtering characteristics of the unit cell designed in Scenario 1 with different numbers of finite periodic arrays are given in Figure 11b. The periodic arrangement of the geometry considered in Scenario 1 significantly increased the passband fluctuation levels, as shown in Figure 11b.