## 3 Transition Design  
The 3D view of the proposed topology for an improved wideband transition from microstrip line to one layer RESIW is shown in Figure 2a. The detailed layout with dimensions of the central (Figure 3a) and ridge layers (Figure 3b) of this transition is shown in Figure 3.  
The transition is based in four elements in order to couple the fields and match the impedances of the input and output transmission lines. Since there is a large difference between the impedances of the microtrip (low impedance) and the RESIW (high impedance), the first element consists on a linear taper at the end of the microstrip line that gradually modifies the impedance of the microstrip line, thus easing the impedance matching with the RESIW. The second element of the transition is a widened RESIW section, placed at the beginning of the RESIW. This section is controlled with the parameters \(a_{c}\) (length) and \(b_{c}\) (width). This widening properly decreases the impedance of the RESIW,  
Figure 1: Cross sectional view of the ESIW and the ridge ESIW with one layer. Figure 2: Structure of the proposed transition from microstrip to RESIW (**a**); and simulated field distribution for the central frequency (\(14\,\mathrm{GHz}\)) (**b**). so that the impedance matching with the microstrip is easier. This widening is manufactured in the central and in the ridge layer (see Figure 3a,b). The third element of the transition is an inductive iris of width \(w_{ti}\), mechanized with rows of metallized via holes, that controls the coupling between the widened microstrip line and the widened RESIW. These three elements were already present in the previous version of the MS-RESIW transition [16] (see Figure 4a). In that previous transition, the matching was completed with the fourth element of the transition, which was a widening of the ridge at the beginning of the RESIW [16]. In this work, instead of using a widening of the ridge, the ridge has no widening and a dielectric taper is used in the central layer in order to better match the discontinuity between dielectric and air. The use of a dielectric taper, never used before in a transition from MS to RESIW, is inspired by the dielectric taper used in the transition from microstrip line to ESIW presented in [2] and improved in [11] (see Figure 4b). However, and conversely to what happens in the transition from MS to ESIW, the exponential dielectric taper provides poor results in the case of a transition from MS to RESIW. For that reason, in this work, we propose the use of the equations of the superellipse [20] for shaping the dielectric taper. This curve allows to explore an endless set of possible transition profiles, using just a few number of parameters and finding the optimum parameter values that provide with an excellent performance. Figure 3: Layout of the microstrip line to RESIW transition proposed in this work. White represents empty spaces, gray represents dielectric body, and dark yellow represents copper surfaces. The equations of the superellipse have been used in several fields of the engineering, especially in architecture [21]. In the microwave field, they were used in [20] for defining a tapering impedance matching profile between two rectangular waveguides of different widths. The superellipse was first discussed in 1818 by [PERSON]. It consists in a generalization of the equations of the ellipse, so that the ellipse is a particular case of the superellipse. The Cartesian equation of superellipse is,  
\[\left(\frac{x}{a}\right)^{m}+\left(\frac{y}{b}\right)^{m}=1 \tag{1}\]  
where \(b\) and \(a\) are the semiaxes of the superellipse and \(m\) is the parameter controlling the overall shape of the ellipse. By properly selecting the value of \(m\), a rectellipse (\(m=4\)), an ellipse (\(m=2\)), a straight line (\(m=1\)), an astroid (\(m=2/3\)), or other shapes can be obtained.