# An In-Line Coaxial-to-Waveguide Transition for Q-Band Single-Feed-Per-Beam Antenna Systems
## 2 Design
Nevertheless, to improve the matching over a wide band, modifications in the back-short of the WG and in the coaxial are needed. Several strategies can be used to achieve this aim, as modifying the WG height \(\mathrm{b}\) in the \(\mathrm{l}_{\mathrm{p}}\) step or the insertion of reactive components. In this work, as shown in Figure 2, the vertical side of the backshort is shortened from \(\mathrm{b}\) to \(\mathrm{b}_{1}\) in order to decrease the wave impedance to match it with the end-launcher and the ridged step, and a sort of rectangular cavity with the same width of the WG and respectively \(\mathrm{l}_{\mathrm{m}}\), \(\mathrm{h}_{\mathrm{m}}\) in depth and height is added. Such a cavity acts as a tuning stub whose reactive behaviour improves the impedance matching in the transition. Moreover, a reactance-series is introduced in the coaxial cable to further compensate the input reactance, realized through a discontinuity in the dielectric insulator. The Teflon is removed at \(\mathrm{l}_{\mathrm{d}}\) distance from the transition to create a capacitance which balances the reactive parasitic effects inside the waveguide (highlighted in green in Figure 2). Such modifications do not increase the transverse section of the transition, maintaining the aim of a compact structure. Finally, the ridged WG must be matched with the hollow WR-22 over the whole required bandwidth. This is realized through a multi-step  
Figure 3: Field distribution in the mode transformation (in magnitude) at two different transverse sections. The white areas are metal filled. With reference to the Figure 2: (**a**) Electric field of the TEM mode on the cross-section at center of the \(\mathrm{l}_{\mathrm{p}}\) long step where the coaxial pin penetrates in the WG; (**b**) Electric field of the \(\mathrm{TE}_{10}\) mode on the cross-section at the center of \(\mathrm{l}_{\mathrm{r}}\) long ridged WG step. Figure 2: Geometry of the coaxial-to-waveguide transition: sectional lateral view. The pin \(\mathrm{l}_{\mathrm{p}}\) long is shorted on a ridge \(\mathrm{h}_{\mathrm{r}}\) high and \(\mathrm{l}_{\mathrm{r}}\) long (in dark grey). The unrridged WG step has a reduced height \(\mathrm{b}_{1}\), and a matching cavity of dimensions (\(\mathrm{h}_{\mathrm{m}}\times\mathrm{l}_{\mathrm{m}}\)) is added along the backshort to improve the matching. The coaxial step highlighted in green indicates the \(\mathrm{l}_{\mathrm{g}}\) long air gap inserted at a distance \(\mathrm{l}_{\mathrm{d}}\) from the transition. transformer based on ridges of progressive descending height. The bandwidth required for the impedance matching is connected with the number of steps in the transformer [40]. Because of the complexity of the geometry, there is not a precise guideline for the sizing of the whole transition, and an analytical model is not trivial to be defined due to the number of parameters involved. Early, the first ridge size is defined in order to suitably fit the coaxial impedance and the electric field distribution and in the same time not to complicate the manufacturing issue (e.g., a too short ridge can make the pin shorting challenging). After this, the backshort and the air gap in the end-launcher are sized together to obtain the desired matching: the first reduce the impedance in the first WG step, where the presence of the coaxial pin avoids the adoption of a ridge, whereas the gap balances the reactive component of the WG input impedance. The \(\mathrm{I_{p}}\) long section is the main part where the modal conversion applies [28]: it consists on a TEM structure with a rectangular outer conductor and a circular inner conductor, the coaxial pin, where the impedance is reduced by resizing the WG height \(\mathrm{b_{1}}\), and shaping the backshort with the cavity, rather than shaping and acting on the cylindrical pin radius, which is a more complicated technological solution, as done in [28].