# Design of a Broadband Transition from a Coaxial Cable to a Reduced-Height Rectangular Waveguide
## 3 Design
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(b/a\) & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \multirow{3}{*}{0.125} & \(a,b,L\) & 19.05, 2.38, 15.00 & \(L_{2},L_{3}\) & 4.19, 11.02 \\ \cline{1-1} \cline{2-5}  & \(a_{4},b_{4},L_{4}\) & 19.77, 2.32, 6.27 & \(D_{1},H_{0},H_{1},S_{1}\) & 1.74, 0.962, 1.10, 9.53 \\ \cline{1-1} \cline{2-5}  & \(a_{1},b_{1},L_{1}\) & 19.50, 2.25, 6.33 & \(D_{2},H_{2},S_{2}\) & 4.63, 0.462, 7.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensions of the proposed transition (unit: mm). Figure 4: Reflection coefficient versus the tuning post dimension and position in millimeters for the transition with \(b/a=0.125\): (**a**) post diameter \(D_{2}\); (**b**) post height \(H_{2}\); (**c**) distance \(L_{2}\) of the post from the disk probe; and (**d**) distance \(S_{2}\) of the post from the side wall. Figure 5: Variation in the reflection coefficient of the transition with \(b/a=0.125\) during a full-parameter optimization process. The final optimized reflection coefficient is shown in a thick blue curve with intermediate results in thin-line graphs of different colors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(b/a\) & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \multirow{3}{*}{0.250} & \(a,b,L\) & 19.05, 4.76, 15.00 & \(L_{2},L_{3}\) & 4.22, 14.57 \\ \cline{2-5}  & \(a_{4},b_{4},L_{4}\) & 19.70, 4.56, 4.15 & \(D_{1},H_{0},H_{1},S_{1}\) & 2.62, 0.916, 2.77, 9.53 \\ \cline{2-5}  & \(a_{1},b_{1},L_{1}\) & 19.90, 4.36, 5.46 & \(D_{2},H_{2},S_{2}\) & 5.42, 0.873, 6.30 \\ \hline \multirow{3}{*}{0.375} & \(a,b,L\) & 19.05, 7.14, 15.00 & \(L_{2},L_{3}\) & 4.43, 10.67 \\ \cline{2-5}  & \(a_{4},b_{4},L_{4}\) & 19.32, 6.80, 7.58 & \(D_{1},H_{0},H_{1},S_{1}\) & 2.32, 1.05, 3.54, 9.53 \\ \cline{2-5}  & \(a_{1},b_{1},L_{1}\) & 19.66, 6.51, 5.99 & \(D_{2},H_{2},S_{2}\) & 3.05, 1.75, 6.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 6: Normalized electric (**a**) and magnetic (**b**) fields inside the transition with \(b/a=0.375\). Figure 7a shows the structure and the simulation results of the proposed transition with \(b/a=0.125\). Figure 7b shows the reflection coefficient at the waveguide port with the TE\({}_{10}\) mode and the transmission coefficient from the output waveguide TE\({}_{10}\) mode port to the coaxial cable TEM port of the CNT. It can be seen that the reflection coefficient \(\mid\)\(S_{22}\)\(\mid\) at the waveguide port is \(<\)\(-\)20 dB at 8.04-19.54 GHz. Without the loss in the transition, the transmission coefficient squared \(\mid\)\(S_{12}\)\(\mid\)\({}^{2}\) is given by 1 \(-\)\(\mid\)\(S_{22}\)\(\mid\)\({}^{2}\) from the law of energy conservation. The reflection coefficient \(\mid\)\(S_{11}\)\(\mid\) at the coaxial cable port is almost identical to \(\mid\)\(S_{22}\)\(\mid\) in the operating frequency range and, thus, is not drawn in Figure 7b. Figure 8 shows the higher-order mode generation in the rectangular waveguide. It was obtained by assuming the output waveguide port is perfectly matched for all nonnegligible modes and by plotting the transmission coefficient from the coaxial cable TEM port to the waveguide TE\({}_{mn}\) and TM\({}_{mn}\) port. Only the transmission coefficients with an appreciable magnitude were included in Figure 8. At the transition's operating frequency range of 8.04-19.54 GHz, the higher-order mode with the largest strength is the TE\({}_{30}\) mode, having a transmission coefficient of \(-\)70.0 dB at 19.54 GHz. Due to the structural symmetry in the \(x\) direction, the generation of the TE\({}_{20}\) mode is suppressed to an extremely low level and is not plotted in Figure 8. With \(a=19.05\) mm and \(b=0.125a=2.38\) mm, the cutoff frequency of the first eight higher-order modes in the output waveguide is 15.74, 23.61, 31.47, 39.34, 62.96, 63.44, 64.89, and 67.23 GHz for TE\({}_{mn}\)/TM\({}_{mn}\) modes with \(mn=\) 20, 30, 40, 50, 01, 11, 21, and 31, respectively. The results for the transition with \(b/a=0.25\) are shown in Figures 9 and 10.