# Path Loss Model for 3.5 GHz and 5.6 GHz Bands in Cascaded Tunnel Environments
## 4 Determination of ELC by RT Simulation  
To simulate the cascaded tunnel, a three-dimensional model composed of a straight tunnel with a length of 600 m and a curved tunnel with a length of 400 m is established. As shown in Figure 10a, \(\mathrm{T_{x}}\) is in the straight tunnel, and the distance of \(\mathrm{T_{x}}\) from the curved tunnel is \(d\). \(\mathrm{R_{x}}\) is placed after every 10 m in the curvature (40 positions in total). To extract the EL generated by the curve, a straight tunnel model with the same dimensions is established, as shown in Figure 10b, and the antenna positions are identical to those used in the cascaded tunnel. The cross-section of both tunnels and the height of the antennas are the same as shown in Figure 6a. The above two models and antenna configurations are imported into Wireless Insite for channel simulations. The simulation parameters are shown in Table 4. According to (2)-(5), the arched tunnel is equivalent to a rectangular tunnel with \(w\) = 4.73 m and \(h\) = 4.23 m. Then, the BP of each frequency band is calculated by using (1), and three different \(d\) are selected for each frequency band using (23), as shown in Table 3.  
From the results of RT simulation, it can be found that the curvature of the cascaded tunnel will produce obvious EL compared with the \(PL_{straight}\) in the straight tunnel when \(f\) = 3.5 GHz, \(d\) = 300 m, and \(R\) = 300 m, as shown in Figure 11a. This phenomenon is due to the transition of the waveguide mechanism when there is a curvature in tunnels. As the direct path is blocked, the enhanced waveguiding mechanism would be taken over by degraded waveguiding, resulting in greater path loss. Taking \(PL_{straight}\) as the reference value, the EL can be extracted and fitted with the direct proportional function in Figure 11b. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(f\) (GHz) & \(d_{1}\) (m) & \(d_{2}\) (m) & \(d_{3}\) (m) & BP (m) \\ \hline
3.5 & 300 & 350 & 400 & 261 \\
5.6 & 450 & 500 & 550 & 417 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: BP and d at 3.5 GHz and 5.6 GHz bands. Figure 10: The model and antennas setting (**a**) Cascaded tunnel; (**b**) straight tunnel. In Figure 11b, EL is approximately proportional to the propagation distance in the curvature, which is consistent with Equation (12). The root mean square error (RMSE) between EL and the fitting line is 1.39 dB, and the slope of the fitting line is the ELC. The ELC and the average RMSE for different radii of curvature (R) are calculated in Table 5.  
For the 3.5 GHz and 5.6 GHz bands, the average RMSE is between 1.35 dB and 2.17 dB, which justifies that EL can be well fitted into a linear function. In addition, the value of \(d\) has little effect on the ELC when \(d\geq\) BP. As R decreases, the ELC evidently increases, which indicates that smaller radius of curvature leads to larger EL at the same distance. This is because more radio waves will be blocked, creating NLOS environments with propagation by reflections only. On the other hand, the ELC decreases around 3 dB/100 m when R increases from 300 m to 900 m, and the ELC decreases around 1 dB/100 m when R increases from 900 m to 1500 m. From the results, the relationship between ELC and R follows an approximate power function, which is consistent with Equation (19). In Figure 12, the ELCs at 5.6 GHz are slightly higher than 3.5 GHz under the same radius of curvature, which means the higher frequencies are more sensitive to the curvature. This finding implies that EL caused by curvature would nullify a lower propagation attenuation rate at higher frequencies, resulting in the increase in total attenuation at higher frequencies in cascaded tunnels. Hence, the influence of frequency and the curvature should be considered together when predicting path loss in cascaded tunnels. Figure 11: (**a**) Comparison of path loss between straight tunnel and cascaded tunnel; (**b**) EL fitted by direct proportional function.