## 4 Conclusions  
This study aimed to investigate the potential of the AFGSM method in analyzing and designing corrugated parallel-plate waveguide unit cells. The AFGSM method made it simple to find the passband and stopband regions of single and glide-symmetric double CPPW periodic structures. This method led to the development of a novel design procedure for filter designs with CPPW structures. We demonstrate that we can design glide-symmetric double CPPW filters using the AFGSM method. The design procedure demonstrates the ability to determine the minimum unit cell size. The proposed method and design procedure yield a bandstop filter characteristic that is more compact and has a high degree of suppression bandwidth. The AFGSM method allows for thorough studies of the bandgap in corrugated parallel-plate waveguides, no matter the mode number, and it also helps in checking initial manufacturing tolerances. Conceptualization, F.C. and A.O.E. ; methodology, F.C. and A.O.E. ; software, F.C. and A.O.E. ; validation, F.C. and A.O.E. ; analysis, F.C. and A.O.E. ; investigation, F.C. and A.O.E. ; resources, F.C. and A.O.E. ; writing--original draft preparation, F.C. ; writing--review and editing, A.O.E. ; visualization, F.C. and A.O.E. ; supervision, A.O.E. ; project administration, A.O.E. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data that support the findings of this study are available from the corresponding author upon request. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} MM-GSM & Mode Matching-Generalized Scattering Matrix \\ BSF & Band Stop Filter \\ HFSS & High-Frequency Structure Simulator \\ CST & Computer Simulation Technology \\ FHMSIW & Folded Half-Mode Substrate Integrated Waveguide \\ SSPP & Spoof Surface Plasmon Polaritons \\ UC & Unit Cell \\ \end{tabular}   
Appendix A Network Representation Explanation of Expanded Version of Equivalent Circuit of Corrugations and Obtaining S-Parameters of Unit Cell with All Definitions  
To find the scattering matrices of all structures belonging to the unit cells given in Figure 2, it is necessary to find the scattering matrix of each block element \(S_{i}\) (\(i=1,2,...,5\)). Since blocks S1, S3, and S5 in Figure 2 are conventional transmission lines, the scattering parameters of a lossless transmission line of length \(l\) are defined as follows:  
\[\left[S^{TL}\right]=\ \ \ \begin{bmatrix}0&e^{-i\beta_{0}l}\\ e^{-i\beta_{0}l}&0\end{bmatrix} . \tag{1}\]  
By replacing \(l\) and \(k_{0}=2\pi f/c\) (where \(f\) and \(c\) are the frequency and speed of light) with the length and phase constant of the block under consideration, the scattering parameters for blocks S1, S3, and S5 can be found. In this case, \(l\) is \(b\) for S1 and S3 in a single-corrugated PPW; in a glide-symmetric double-corrugated PPW, it is \((p-m-a)/2\) for S1 and S5 and \((m-a)\) for S3. S2 and S4 circuits play a crucial role in determining the scattering parameters of the unit cells in the corrugated PPW structures under study. To clarify this situation, the open forms of the \(\pi\) circuits (S2 and S4) are given in Figure 1,b, with series stubs reduced in Figure 2.  
The reactive elements indicated in Figure A1a,b are located at the junctions to account for higher-order mode interactions [36]. The corrugations consist of a series of stubs terminated with a short circuit. Impedance expressions (\(Z_{A}\) and \(Z_{B}\)) for the reduced forms of the \(\pi\) circuits mentioned in this work are given. Figure A1 presents the derivation of \(Z_{A}\) and \(Z_{B}\) in a sequentially numbered format. In step 1, \(Y_{A}\) is first derived using the provided equality \(Y_{A}=jB_{a}\), followed by the calculation of \(Z_{A}\) as \(Z_{A}=1/Y_{A}\)[49].