# Computational Design of a Broadband In-Line Coaxial-to-Rectangular Waveguide Transition
## 3 Design Examples
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline \(a\), \(b\), \(L\) & 19.05, 8.06–9.53, 15.01 & \(F\), \(D\) & 2.17, 5.60 \\ \hline \(a_{1}\), \(b_{1}\), \(L_{1}\) & 19.47, 6.57–8.06, 15.25 & \(t\), \(G\), \(R\) & 1.81, 1.21, 0.90 \\ \hline \(a_{2}\), \(b_{2}\), \(L_{2}\), \(L_{3}\) & 20.58, 6.57, 1.89, 20.05 & Ridge width/height & 3.23/0.37, 1.98, 3.86, 5.31 \\ \hline Coaxial connector & 1.27/4.11, \(\epsilon_{r}\) = 2.1 (SMA) & Probe hole & Dia. = 2.95, Len. = 1.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensions of the designed WR75 transition. Figure 8: Reflection coefficients during an optimization of the designed WR75 transition. The curve in red is the final reflection coefficient. Figure 7: Higher-order modes in a transition with a linear taper of zero slope. In the final optimized reflection coefficient, in Figure 8, we observe small ripples at 17-20 GHz, larger ripples in at 20-23 GHz, and a spike at 23.15 GHz. This is caused by higher-order modes excited in the mode-converter waveguide \(W_{2}\), where the stepped-ridge transformer acts as an obstacle giving rise to higher-order modes. This can more clearly be understood with the help of Figure 9, where the transmission coefficients of higher-order modes are plotted. At 17-20 GHz, the higher-order mode level is about \(-\)50 dB, while at 20-23 GHz, it steadily increases from \(-\)50 dB to \(-\)20 dB. At 23.15 GHz, all three higher-order modes, namely TM\({}_{11}\), TE\({}_{30}\), and TE\({}_{11}\) have a local peak, causing a large spike in the reflection coefficient of the TE\({}_{11}\) mode. Figure 9 shows the transmission coefficient between the coaxial TEM mode and the rectangular waveguide high-order modes. Analysis of the higher-order modes is important in the design of the proposed transition. In Table 3, we show the cutoff frequency of the TE\({}_{10}\), TE\({}_{30}\), and TE\({}_{11}\) modes in the designed WR75 transition. The lower frequency limit of the transition is determined by the TE\({}_{10}\) cutoff frequency 7.87 GHz (\(f_{10}\)) of the output waveguide \(W\). In Figure 9, the spikes at 23.10 GHz are at the \(-\)13.7 dB level caused by the TM\({}_{11}\) mode, at the \(-\)17.1 dB level due to the TE\({}_{30}\) mode, and at the \(-\)20.0 dB level due to the TE\({}_{11}\) mode. We note that the cutoff of the TE\({}_{11}\), TM\({}_{11}\), and TE\({}_{30}\) modes occurs at the same frequency of 23.10 GHz, as predicted by the design theory. We note in Figure 9 that at 8.14-19.40 GHz, the TE\({}_{12}\) mode is the highest, ranging from \(-\)51.0 dB to \(-\)41.4 dB except for a TM\({}_{11}\)-mode spike of \(-\)44.6 dB at 17.7 GHz. At 19.40- 23.00 GHz, the TM\({}_{11}\) mode is the largest (\(-\)51.0 dB to \(-\)25.0 dB). In Figure 10, we also observe that the TE\({}_{11}\), TM\({}_{11}\), and TE\({}_{30}\) modes steadily increase at 20-23 GHz from about \(-\)50 dB to \(-\)25 dB. This is in contrast to the transition without a linear taper, where the higher-order mode level is less than \(-\)47.4 dB in the operating frequency range. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Waveguide**} & **Dimension** & \multicolumn{2}{c}{**TE\({}_{10}\)-Mode**} & \multicolumn{2}{c}{**TE\({}_{30}\)-Mode**} & \multicolumn{2}{c}{**TE\({}_{11}\)/TM\({}_{11}\)-Mode**} \\  & \(\mathbf{a\times b\) (mm)} & **Cutoff Frequency** & **Cutoff Frequency** & **Cutoff Frequency** \\  & & \(\mathbf{f_{10}\) (GHz)} & \(\mathbf{f_{30}\) (GHz)} & \(\mathbf{f_{11}\) (GHz)} \\ \hline Output Waveguide \(W\) & 19.05 \(\times\) (8.06–9.53) & 7.87 & 23.61 & 17.59–20.18 \\ \hline Matching Waveguide \(W_{1}\) & 19.47 \(\times\) (6.57–8.06) & 7.70 & 23.10 & 20.13–24.08 \\ \hline Converter Waveguide \(W_{2}\) & 20.58 \(\times\) 6.57 & 7.28 & 21.85 & 23.95 \\ \hline \end{tabular}
\end{table} 
Table 3: Waveguide modes and their cutoff frequencies in the designed WR75 transition. Figure 9: Higher-order modes in the designed WR75 transition.