## Appendix A Study of Possible Geometries of a RESIW Crosssection  
In this Appendix, an exhaustive study of the possible geometries of the RESIW is presented. Figure 11 shows the RESIWs transversal section that have been studied. Figure 10: Comparison of measurements and simulations of the back-to-back transition prototype. In the ridge waveguide, the cutoff frequencies of the fundamental mode \(TE_{10}\) (\(f_{c10}\)) and the first higher order mode \(TE_{20}\) (\(f_{c20}\)) can be obtained by solving the following transcendental equations [22],  
\[\cot\left(\frac{2\pi f_{c_{10}}\left(\frac{a-w_{r}}{2}\right)}{c}\right)-\frac{b }{d}\tan\left(\frac{\pi w_{r}f_{c_{10}}}{c}\right)-\frac{B}{Y_{01}}=0 \tag{1}\]  
\[\cot\left(\frac{2\pi f_{c_{20}}\left(\frac{a-w_{r}}{2}\right)}{c}\right)+\frac {b}{d}\cot\left(\frac{\pi w_{r}f_{c_{20}}}{c}\right)-\frac{B}{Y_{01}}=0 \tag{2}\]  
where \(c\) is the light velocity; \(a\) and \(b\) are the width and height of the waveguide, respectively; and \(d=b-h_{r}\) is the distance between the top ridge layer and top wall of waveguide (see Figure 1). The term \(B/Y_{01}\) is the susceptance that models the gap, and can be approximated as [17]  
\[\frac{B}{Y_{01}}\approx 4\left(\frac{b}{d}\right)\left(\frac{af_{c}}{c}\right) \ln\csc\left(\frac{\pi d}{2b}\right) \tag{3}\]  
where \(f_{c}\) is \(f_{c10}\) for solving Equation (1) and \(f_{c20}\) for solving Equation (2). Equations (1) and (2) can be used to obtain \(f_{c10}\) and \(f_{c20}\) as a function of the width of the ridge, \(w_{r}\), which is the only design parameter in a RESIW that we can use in order to maximize the usable bandwidth (\(f_{c20}-f_{c10}\)). Figure 2 shows both cutoff frequencies \(f_{c10}\) and \(f_{c20}\) as a function of \(w_{r}\) for a RESIW manufactured with Rogers 4003C substrates of height \(h=0.813\) mm, permittivity \(\epsilon_{r}=3.66\) and width \(a=15.7988\) mm. The cutoff frequencies were computed both analytically with Equations (1) and (2) and numerically with the commercial software Computer Simulation Technology (CST). The results are shown for a RESIW with one, two and three ridge layers (\(n=\)1, 2 and 3), and they are compared with the cutoff frequencies of an ESIW (no ridge layers). It can be observed, in the first place, that there is a very good agreement between analytical and simulated results, which validates the accuracy of Equations (A1) and (A2). Besides, it can be concluded that the higher bandwidth (maximum distance between \(f_{c10}\) and \(f_{c20}\)) is achieved with \(w_{r}/a\simeq 0.28\). Using this value of \(w_{r}\) as a starting point, and after optimization, optimum values of \(w_{r}\) that maximize the bandwidth were obtained for the the RESIW with one, two, and three ridge layers, and the values of the cutoff frequencies for these optimum values of \(w_{r}\) are shown in Table A1. The results depicted in Table A1 show that the bandwidth increases with the number of ridge layers. Using one ridge layer increases the bandwidth by 38% compared with the ESIW with no ridge layer. However, using two ridge layers supposes an increase in the bandwidth of only a 15% when compared with one ridge layer. Using three ridge layers increases the bandwidth by 7% compared with two ridge layers. Taking into account that increasing the number of ridge layers increases the volume,weight, cost, and manufacturing complexity, for most wideband applications, the best compromise between bandwidth and ease of manufacturing will be the use of one ridge layer. To test whether the usable bandwidth of the RESIW with one ridge layer can still be improved by using other unconventional geometries, the trapezoidal and two rectangular ridge geometries shown in Figure 14 were also studied. Table 15 shows the cutoff frequencies and the absolute bandwidth of these geometries compared with the rectangular cross section single layer ridge.