# Computational Design of a Broadband In-Line Coaxial-to-Rectangular Waveguide Transition
## 3 Design Examples
Figure 10 shows the simulated performance of the designed WR75 transition in a decibel scale and in a natural scale as well. The reflection coefficient of the TE\({}_{10}\) mode is plotted along with the transmission coefficient between the coaxial TEM mode and the output waveguide TE\({}_{10}\) mode. The reflection coefficient (\(\mid\!\Gamma\!\mid\)) is less than \(-22.6\) dB at 8.14-23.0 GHz (2.83:1 bandwidth). The transmission coefficient is determined via the relationship \(1-\mid\!\Gamma\!\mid^{2}\). In Figure 10a, the start frequency for the reflection coefficient \(<-20\) dB is 8.09 GHz (=1.028\(f_{10}\)). The transition maintains a reflection coefficient \(<-22.6\) dB from 8.14 GHz to 23.00 GHz (=0.974\(f_{30}\)). The theoretical upper frequency limit of the transition is determined by the level of the higher-order TE\({}_{11}\) and TE\({}_{30}\) modes generated in the waveguides \(W\), \(W_{1}\), and \(W_{2}\). A spike in the reflection coefficient is generated at 23.1 GHz, where the TE\({}_{11}\), TM\({}_{11}\), and TE\({}_{30}\) modes have a transmission peak all at the same time, as shown in Figure 9. The transition operates from slightly above the TE\({}_{10}\) mode cutoff to slightly below the TE\({}_{30}\) cutoff. The resultant bandwidth is 2.83:1 or 94.3% of the theoretical limit of 3:1. In Table 3, we note that the cutoff frequency of the TE\({}_{11}\), TM\({}_{11}\) and TE\({}_{30}\) modes ranges from 20.13 GHz to 24.08 GHz in the matching waveguide \(W_{1}\) and from 17.59 GHz to 23.61 GHz in the output waveguide \(W\). In Figure 10, there are low-level spikes in the higher-order mode transmission coefficients: \(-44.5\) dB at 17.7 GHz due to the TM\({}_{11}\) mode, \(-42.4\) dB at 20.4 GHz due to the TM\({}_{11}\) mode, and \(-46.1\) dB at 20.5 GHz due to the TE\({}_{11}\) mode. These spike frequencies closely correlate with the cutoff frequency of the TE\({}_{11}\) and TM\({}_{11}\) modes in the matching and output waveguides. Figure 10: Reflection and transmission coefficients of the designed WR75 transition: (**a**) in dB scale and (**b**) in natural unit. During the optimization of the WR75 transition, we observed that the resonance spikes of the TE\({}_{11}\) and TM\({}_{11}\) modes can be generated between 20 GHz and 22 GHz. It was found that this is not caused by the steps in the waveguide broad wall. This can be checked by analyzing the structure with the waveguides \(W\), \(W_{1}\), and \(W_{2}\) only (i.e., with the structure with \(S\), \(B\), \(P\), and \(X\) in Figure 3 removed). It is caused by the stepped-ridge transformer \(S\) in the converter waveguide \(W_{2}\). Careful efforts are required for a transition design having a bandwidth approaching the theoretical limit of 3:1. It is because the TE\({}_{11}\) and TM\({}_{11}\) modes can be generated by the stepped-ridge transformer in the converter waveguide and then transmitted to the matching and output waveguides. Thus, the design effort should be focused on the suppression of the TE\({}_{11}\) and TM\({}_{11}\) modes while achieving a low reflection coefficient over a broad frequency range. The designed WR75 transition was obtained by simultaneously optimizing for both the low reflection coefficient and the low level of higher-order modes. To achieve a much smaller reflection coefficient in the range of \(-\)30 dB with the proposed transition, a further investigation is necessary that involves a re-design of the stepped-ridge transformer, including an increase in the number of steps. On the other hand, it may even require a different approach to the TEM-to-TE\({}_{10}\) mode transformer design. A literature survey revealed that a design with a reflection coefficient in the level of \(-\)30 dB over such a broad bandwidth seems to be rare if not impossible. The problem of designing a very-low-reflection broadband transition is left as a future research topic.