# An In-Line Coaxial-to-Waveguide Transition for Q-Band Single-Feed-Per-Beam Antenna Systems
## 2 Design
In the same time, as aforementioned, the field in such a modal conversion step is concentrated the gap between the coaxial pin and the upper wall, by simulating the modal distribution in the downstream R-WG. With reference to Figure 2, the pin is \(\mathrm{I_{p}}=0.4\) mm long and g = 0.205 mm distant from the upper wall of the WG. The ridge where the pin is shorted keeps the same hollow space above, so it results \(\mathrm{h_{r}}=2.64\) mm high, \(\mathrm{I_{r}}=1.2\) mm long and w = 1.1 mm wide. The hollow step inside the coaxial cable is \(\mathrm{I_{g}}=0.52\) mm long and it is placed \(\mathrm{I_{d}}=1.2\) mm distant from the pin. The Smith Chart in Figure 4 shows the effect of the air gap insertion. The backshort height is reduced to \(\mathrm{b_{1}}=2.56\) mm in the step where the pin enters inside the WG, and a cavity \(\mathrm{h_{m}}=1.28\) mm high and \(\mathrm{l_{m}}=0.94\) mm deep is added behind it. The Chebyshev impedance transformer has been realized in 5 steps, which have proven to be the best trade-off between the RL performance and the specification of small transition. All the ridges have the same width of the first (1.1 mm). The requirement of high RL at such frequencies requires very precise size for the transformer steps, more than the values that can be obtained by the approximated formula for R-WGs [39]. Moreover, also the effects of the reactive phenomena due to the discontinuities between each couple of ridges must be traded by correcting the ridge lengths. To overcome these two issues, the Chebyshev impedance transformer has been optimized through the PSO algorithm, widely  
Figure 4: Effect of the air gap in the coaxial cable: the return loss with the air gap (blue line) and without (red). used in electromagnetic problems [41; 42], in order to determine the lengths and the heights of the ridges which minimize the RL seen at the highest ridge section. The optimization procedure is based on the combined operation of MATLAB 2015 and a Finite Element Method (FEM) 3D electromagnetic solver (Ansys Electronics Desktop 2016.2), respectively in charge of the optimization algorithm script definition and of the electromagnetic performance evaluation, in order to minimize the overall RL in the required frequency band. A conceptual scheme of this process is explained in Figure 5. The algorithm considers as swarm positions the deviation of the R-WG geometrical parameters (in this case, the ridge heights and lengths) from the theoretical values calculated via the formula available in [39], then calculates the objective functions (which accounts for a maximum RL over the whole Q-band) through the FEM solver and iterates till convergence is reached. The working principle of such an optimization process has been explained in details in [33] and the optimized values of the ridges' size (displayed in Figure 6) are reported (in millimeters) in the Table 1.  
Figure 5: Summary of the PSO-based optimization algorithm for the Chebyshev transformer [33]. Figure 6: Ridge cascade structure: side view. The first ridge (of size \(\text{h}_{r}\), \(\text{l}_{r}\)) is the same as in Figure 2, where the coaxial pin is shorted. The remaining ridges make up the Chebyshev impedance transformer, whose size \(\text{h}_{i}\), \(\text{l}_{i}\), \(\text{i}=1,\ldots,5\) are optimized via the PSO algorithm.