# Meander Structure Analysis Techniques Using Artificial Neural Networks
## 3 Results
### Results Obtained with Feedforward Neural Network  
The results for \(t_{\text{d}}\), \(S_{11}\), and \(S_{21}\) are presented in Figure 6. The prediction results for the \(t_{\text{d}}\) characteristic in four meander structures, labeled as MS1, MS2, MS3, and MS4, were obtained with the (6-9-11-1) structure for the FFNN network, and they are presented in Figure 6. The prediction results for the \(t_{\text{d}}\) characteristic were compared with the calculation results when using the MoM method in the Sonnet\(\copyright\) software package Version 18. First of all, it should be mentioned that the \(t_{\text{d}}\) characteristic is an important parameter for meander structures in the field of signal synchronization indicating the possible signal \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Title of** & \(L_{\text{c}}\) **(mm)** & \(I\) **(mm)** & \(2a\) **(mm)** & \(c\) **(mm)** & \(N\) \\ \hline MS1 & 0.4 & 0.2 & 10 & 6 & 14 \\ MS2 & 0.55 & 0.45 & 10 & 6 & 14 \\ MS3 & 0.9 & 0.7 & 10 & 6 & 14 \\ MS4 & 0.6 & 1.45 & 10 & 6 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Configurations of meander structures which were used for result visualization. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(t_{\text{d}}\) **(ps)** & \(S_{21}\) **(dB)** & \(S_{11}\) **(dB)** \\ \hline LSBoost & 5.1 \(\pm\) 7.2 & 0.38 \(\pm\) 0.44 & 1.94 \(\pm\) 1.89 \\ FFNN & 3.3 \(\pm\) 1.5 & 0.64 \(\pm\) 0.42 & 2.47 \(\pm\) 1.55 \\ TDNN when \(N_{\text{d}}=8\) & \(\mathbf{0.9\pm 0.64}\) & \(\mathbf{0.11\pm 0.12}\) & 1.63 \(\pm\) 1.71 \\ CNN & 27.5 \(\pm\) 3.5 & 0.44 \(\pm\) 0.46 & \(\mathbf{1.36\pm 0.72\ dB}\) \\ \hline \hline \end{tabular} The text in bold represents the best neural network option for prediction of \(t_{\text{d}}\), \(S_{11}\) and \(S_{21}\) characteristics.  
\end{table} 
Table 2: Prediction accuracy of \(t_{\text{d}}\), \(S_{21}\), and \(S_{11}\), represented by MAE with SD values. Figure 6: The **(a)**\(t_{\text{d}}\), **(b)**\(S_{21}\), and **(c)**\(S_{11}\) of MS1 and MS2 and **(d)**\(S_{11}\) of MS3 and MS4 when the (6-9-11-1) structure for an FFNN was used for \(t_{\text{d}}\) prediction and the (6-9-16-1) structure for an FFNN was used for \(S_{21}\) and \(S_{11}\) prediction. delay in the operating frequency range. The delay time characteristic depends on the frequency. Therefore, it is important to perform the analysis across the entire selected operating frequency range. The comparison showed good agreement between the results calculated with Sonnet(r) and predicted with an FFNN network. The average MAE between the predicted and simulated results of the 74 tested MSs did not exceed \(3.3\pm 1.5\) ps (Table 2), or approximately 0.3% when taking into account that the desired delay time was to the order of 1 ns. The average MAE of \(t_{\rm d}\) was even smaller if only the bandwidth from 0 until approximately 3 GHz was analyzed, being equal to \(3\pm 1.5\) ps. Meanwhile, the \(t_{\rm d}\) values for four MSs are presented in Figure 6a for visual analysis. For example, the maximum peak error value of MS1 was equal to 20 ps at 5.4 GHz. The \(S_{21}\) parameter represents information about a meander structure's transmission efficiency, frequency, and phase responses. It is extremely important to understand the frequency-dependent behavior of meander structures in order to ensure optimal performance of microwave devices. The prediction results for \(S_{21}\) were obtained with the (6-9-16-1) structure for the FFNN network. The average MAE of \(S_{21}\) did not exceed \(0.64\pm 0.42\) dB in the overall investigated 0-10 GHz frequency range (Table 2). The cutoff frequency of the bandwidth varied depending on the meander structure (MS1 = 3.78 GHz, MS2 = 3.65 GHz, MS3 = 3.05 GHz, and MS4 = 2.9 GHz; see Figure 6b). For example, the maximum peak error value of MS1 was equal to 5.36 dB at 9.15 GHz. The largest average MAE was obtained by predicting the \(S_{11}\) results.