## 4 Results  
To validate the transition, a back-to-back transition from a 50 \(\Omega\) microstrip line to a RESIW with one ridge layer was designed, manufactured, and measured. A Rogers 4003C substrate with \(h_{r}=0.813\) mm, permittivity \(\epsilon_{r}=3.55\), and metal thickness \(t=35\) um was used. The transition presents the same width of the standard WR-62 rectangular waveguide (\(a=15.7988\) mm). The expressions in Table 1 were applied to obtain initial values for the design parameters. Since the initial values were good enough, the optimum values were obtained after a quick and simple optimization process with the Neder-Mead Simplex algorithm of CST and the goal of maximizing the return loss. Table 2 shows the initial values of the design parameters and the final optimum values. It also shows the dimensions that are fixed and that do not need to be optimized. Figure 6 compares the simulated (with CST) reflection coefficient of the new transition presented here and the previous version presented in [16]. To assess the validity of the initial values of the design parameters, the reflection coefficient with these initial values is also plotted. It can be observed that the initial values provide a minimum return loss of about 12 dB in the whole band, which proves that the initial vales of Table 1 are a very good approximation to the final optimum values. Comparing the optimized transition proposed here and the previous transition presented in [16], it can be observed that the fractional bandwidth (calculated as the frequency range in which the return losses are greater than 20 dB divided by the central frequency, 14 GHz) is 82% with the previous transition of  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**\(w_{n}\)** & \(w_{t}\) & \(l_{t}\) & \(a_{c}\) & \(b_{c}\) & \(w_{sell}\) & \(l_{sell}\) & \(l_{2}\) & \(p\) & \(q\) \\ \hline
0.7\(a\) & 0.2\(w_{i}\) & \(\frac{\lambda_{m}(f_{0})}{4}\) & \(\frac{\lambda_{p}(fHerraiz et al. [16] and 87% with the new transition, which supposes an increment of 5%. It can also be observed that the optimized new transition presents a return losses value greater than 26 dB from 8.7 to 20.5 GHz, which is a very good matching in a large bandwidth, thus proving its good performance in terms of simulated results. The simulated insertion losses of the previous [16] and new transition are compared in Figure 7. Lossy metal and lossy dielectric are considered in the simulation with CST. It can be observed that the insertion losses of the new transition are lower than 0.6 dB in the whole band, whereas the insertions losses of the previous transition are greater than 1.2 dB. Therefore, the simulated results of the new version of the transition are better in terms of both insertion and return losses than the previous version of [PERSON] et al. [16]. Figure 8 compares the return losses of the the back-to-back transition from microstrip to ESIW without ridge [11] and the new back-to-back transition from microstrip to RESIW presented here. There is an increment of 103% in the fractional bandwidth (return losses greater than 20 dB) with the new transition when compared with the transition from MS to ESIW without ridge, whose fractional bandwidth is 42%. Figure 6: Return loss comparison between previous [16] and new version of the back-to-back transition from microstrip to RESIW. For the new transition, results of the design with initial values of the parameters and of the design with final optimized values are plotted. Figure 7: Insertion loss comparison between previous [16] and new back-to-back transition from microstrip to RESIW. The manufactured back-to-back prototype with the final design parameters of Table 2 is shown in Figure 9. Standard planar circuit manufacturing techniques (drilling, milling, and electrodeposition) were used to manufacture this prototype. Figure 10 compares the simulated and measured scattering parameters of the back-to-back transition prototype.