## 3 Results and Discussion
### Design of Unit Cell and Cascade Connection Analysis of Corrugated PPW Structures  
The design phase begins with the selection of an appropriate unit cell model. We select the initial parameters \(h_{1}=1\) mm, \(h_{2}=1.8\) mm, \(a=0.8\)\(h_{2}\), and \(b=2.2\)\(h_{2}\) for the air-filled single-corrugated PPW model. This selection implements the design procedure as outlined in the initial step. At this point, the design of the unit cell transitions to the determination of the critical parameters of the design geometry. Figure 9 illustrates the relationship between the geometrical parameters of the AFGSM functions and the zero crossings across various frequencies. In the second step, the design procedure maintains the constrained design spaces (with respect to \(h_{1}\) and \(h_{2}\) for \(h_{1}=1\) mm and \(h_{2}=1.8\) mm) while restricting \(a\) and \(b\) to specific ranges, as illustrated in Figure 9. The variations given in this figure provide a solution for obtaining the parameters corresponding to the targeted stopband bandwidth for the parameters under investigation. Figure 8: **(a)** CST-FD dispersion diagram. \(X_{+}^{full}\), \(X_{-}^{full}\) frequency behavior of the dominant mode of a single CPPW unit cell, and the dispersion diagram of the equivalent circuit model geometry under multimode excitation. **(b)** Tolerance analysis of glide-symmetric DCPPW unit cell with \(\Delta a\) variation using full-wave simulator assisted \(X_{+}^{full}\), \(X_{-}^{full}\). Figure 9: Frequency dependence of bandgap on \(a/h_{2}\) and \(b/h_{2}\) computed via AFGSM functions \(X_{+}\) and \(X_{-}\). Step 3 investigates the variation in \(X_{+}\) and \(X_{-}\) functions for the dimensions in the limited design space obtained in step 2. Figure 9 illustrates that the \(X_{+}\) and \(X_{-}\) functions for the lines \(b=2\)\(h_{2}\) and \(a=0.6\)\(h_{2}\) possess a stopband that satisfies the design requirements, as depicted for X and Y point pairs. The AFGSM method enables the initial selection and simultaneous modelling of specific geometric parameters within the frequency range that meets design criteria, offering the designer precise alternatives for determining the optimal geometry of the unit cell, as demonstrated in Figure 9. In the final step, Figure 10a shows the dispersion diagram of the designed unit cell, satisfying the suppression-level design criterion. We obtained a unit cell design with a stopband of 4.06 GHz bandwidth and a center frequency of 16.51 GHz. To test the band properties, we connected unit cells back to back in varying numbers, as illustrated in Figure 10b. Figure 10b shows that the full-wave electromagnetic solver results are quite compatible. The \(|S_{21}|\)-dB results of the EC model for the cascade-connected single-corrugated PPW unit cells exhibit a less deep stopband region in contrast to the \(|S_{21}|\)-dB results from the full-wave electromagnetic model of the structure. The EC model is inadequate for accurately representing the discontinuities in the cross-sections for this structure. Discontinuities in the sections result in the excitation of high-order modes. Only the reactive components on the junction planes receive evaluation for high-order mode interactions in the circuit model [36]. Mode-matching methods may be used on these structures to obtain results comparable to full-wave electromagnetic simulation outcomes [31]. As it is well-known in periodic structures, increasing the number of unit cells leads to a deeper stopband in single CPPWs. In this example, we ensured adequate design requirements by increasing the number of unit cells (\(N\)) to 10. For glide-symmetric double CPPW structures, we set the parameters to \(a=1\) mm, \(h_{1}=0.5\) mm, \(h_{2}=h_{3}=1\) mm, and \(b=12.4\) mm. This selection implements the design procedure as outlined in the initial step. We limited the \(a\) and \(h_{2}=h_{3}\) parameters to 1-2.5 mm and \(b\) to 12.4-13.6 mm at intervals of 0.1 mm by creating multiple design spaces and preserving the glide-symmetry case.