## 4 Bending Analysis
### Effect of Bending on Resonant Frequencies  
The results obtained from the experimental verifications of the \(S\)-parameters inside the chamber were analyzed with the shift in the resonant frequency for the various bending levels provided in Table 3, which shows that the bending or curvature with bend states of 27 and 14 mm on polymer-based antennas generate different impacts on their FS. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Substrates** & **PET** & **PTFE** & **PVC** \\ \hline \multirow{6}{*}{Operating at 2.45 GHz} & Flat & 2.426 & 2.438 & 2.417 \\ \cline{2-5}  & 27 mm & 2.412 & 2.469 & 2.394 \\ \cline{2-5}  & Shift (\%) & \(-\)0.58 & 1.25 & \(-\)0.96 \\ \cline{2-5}  & 14 mm & 2.402 & 2.484 & 2.366 \\ \cline{2-5}  & Shift (\%) & \(-\)0.99 & 1.85 & \(-\)3.42 \\ \hline \multirow{6}{*}{Operating at 4.25 GHz} & Flat & 4.312 & 4.381 & 4.267 \\ \cline{2-5}  & 27 mm & 4.442 & 4.392 & 4.294 \\ \cline{1-1} \cline{2-5}  & Shift (\%) & 2.92 & 0.25 & 0.62 \\ \cline{1-1} \cline{2-5}  & 14 mm & 4.468 & 4.453 & 4.366 \\ \cline{1-1} \cline{2-5}  & Shift (\%) & 3.49 & 1.61 & 2.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Percentage (%) FS in resonant frequencies for flexible antennas on PET, PTFE, and PVC substrates at 3 operating frequencies with bend conditions of 27 and 14 mm. Figure 12: Cylindrical polystyrene foam to produce flexible antenna bend (**a**) at 27 mm, (**b**) at 14 mm, (**c**) connected to VNA at 14 mm. As observed on the flexible antennas operating at 2.45 GHz, which is a frequency in the ISM-band used for domestic purposes that lies in the first group ranging from 2.2 to 2.5 GHz, the Teflon-based antennas undergo the highest percentage shift in frequency, of approx. 1.25%, towards the highest components of the resonant frequency for a bending state of 27 mm. This result is matched with our investigation in [1]. In contrast, the PET \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Substrates** & **PET** & **PTFE** & **PVC** \\ \hline \multirow{5}{*}{Resonant Frequency (GHz)} & Flat & 7.387 & 7.443 & 7.507 \\ \cline{2-5}  & & 27 mm & 7.425 & 7.429 & 7.541 \\ \cline{2-5}  & Operating at 7.45 GHz & Shift (\%) & 0.51 & \(-\)0.01 & 0.45 \\ \cline{2-5}  & & 14 mm & 7.464 & 7.421 & 7.658 \\ \cline{2-5}  & Shift (\%) & 1.03 & \(-\)0.29 & 1.97 \\ \hline \end{tabular} The negative sign with the frequency shift (\%) indicates that the shift occurs towards the lowest components of the frequency.  
\end{table} 
Table 3: _Cont._  
Figure 13: Operating at 2.45 GHz, reflection coefficients of E-shaped flexible polymer substrate antennas on PET, Teflon, and PVC substrates non-bent and with bending: (**a**) 27 mm, (**b**) 14 mm. and PVC-based flexible antennas possessed less than 1% of frequency shift towards the lowest components of the operating frequency, indicated by the negative sign with the percentage shift. In addition, at the high level of the radial curvature up to 14 mm, the PET substrate antenna yields very little deviation from the resonant frequency compared to the flat state, which is about 0.99% towards the lowest component; see Figure 13.  
For the second operating frequency of 4.25 GHz which lies in the C-band suitable for the WiMAX frequency, the \(S\)-parameters of the flexible microstrip patch antennas reveal that Teflon is much less affected by bending up to 14 mm, providing a 0.25% and 1.61% frequency shift toward the highest components for the bending states of 27 and 14 mm, respectively; see Table 3 and Figure 14. In the first part of this study, in the analysis for the operating frequency range of 2.5-5.0 GHz, PTFE-based flexible antennas were also ascertained to be more efficient by providing 4% of the shift on average. Hence, for this operating range, Teflon is a good candidate as a flexible substrate in terms of the deviation from the resonant frequency of the antenna.