# Computational Design of a Broadband In-Line Coaxial-to-Rectangular Waveguide Transition
## 2 Design Approach
This can be achieved by reducing the narrow-wall height of the mode-converter waveguide \(W_{2}\) so that the TE\({}_{11}\) cutoff of \(W_{2}\) coincides with the TE\({}_{30}\) cutoff of \(W_{2}\):  
\[f_{11}=f_{30} \tag{3}\]  
Then, the narrow-wall height \(b_{2}\) of \(W_{2}\) can be obtained from the following:  
\[(c/2)\left[(1/a_{2})^{2}+(1/b_{2})^{2}\right]^{1/2}=3c/(2a_{2}) \tag{4}\]  
or  
\[b_{2}=a_{2}/\sqrt{8}=0.354a_{2} \tag{5}\]  
With a given waveguide broad-wall width \(a_{2}\), the TE\({}_{30}\)-mode cutoff frequency \(f_{30}\) is fixed, and the TE\({}_{11}\)-mode cutoff frequency \(f_{11}\) can be increased by reducing the waveguide narrow-wall height \(b_{2}\). Making \(f_{11}\) larger than \(f_{30}\) offers no gain since the frequency upper limit will still be \(f_{30}\), and the height difference between the output and converter waveguides will be increased, resulting in an increased mismatch between \(W_{2}\) and \(W\). The broad-wall width of the converter waveguide is chosen to satisfy the following condition:  
\[f_{10}=c/(2a) \tag{2a}\]0  
where \(\Delta\) is a design parameter in the range of 0.04-0.08, and \(a\) is the broad-wall width of the output waveguide \(W\). Equation (6) ensures a transition to operate close to the TE\({}_{10}\)-mode cutoff of the output waveguide. The larger \(\Delta\), the closer to the TE\({}_{10}\) cutoff the start frequency of operation. Large values of \(\Delta\) will increase the reflection at the waveguide width step discontinuity, and more than one section of the matching waveguide may be required. From the foregoing discussion, we find that the proposed transition will work in the following frequency range:  
\[f_{10}=c/(2a) \tag{2a}\]1  
where  
\[f_{10}=c/(2a) \tag{2a}\]2  
The 10 symbol in the subscript denotes the cutoff frequency of the fundamental TE\({}_{10}\) mode. The symbols \(o\), \(m\), and \(c\) refer to the output, matching, and converter waveguides, respectively. The frequency upper limit\(f_{2}\) is determined by the level of the higher-order TE\({}_{11}\) and TE\({}_{30}\) modes generated in the transition. The frequency \(f_{2}\) is more or less determined by the cutoff frequency of TE\({}_{11}\) and TE\({}_{30}\) modes in the waveguides \(W\), \(W_{1}\), and \(W_{2}\). Of course, the bandwidth of the transition is also influenced by the performance of the TEM-to-TE\({}_{10}\) mode transformer, which is a stepped-ridge transformer in our case. The impedance matching waveguide \(W_{1}\) is employed to match the impedance of \(W\) with that of \(W_{2}\). The broad-wall width \(a_{1}\) of \(W_{1}\) is between \(a\) and \(a_{2}\). The width \(a_{1}\), height \(b_{1}\), and length \(L_{1}\) of \(W_{1}\) are determined using an automatic optimization function of the simulation tool. The difference in the height of \(W\) and \(W_{2}\) can be matched in the form of a single- or multi-step transformer or in the form of a continuously tapered transformer. It is important to note that the TE\({}_{11}\) mode should not be generated in the region between the output waveguide \(W\) and the converter waveguide \(W_{2}\) in the operating frequency range of the transition. In fact, during the parametric analysis and the optimization, we observed that the TE\({}_{11}\) mode can be generated in the converter waveguide \(W_{2}\) due to the presence of the stepped-ridge transformer, indicating that a careful design is re quired. In the next section, we apply the foregoing design approach to the design of two coaxial-to-WR75 transitions having a bandwidth exceeding 2.8:1.