# Computational Design of a Broadband In-Line Coaxial-to-Rectangular Waveguide Transition
## 2 Design Approach  
There are many commercial products of the coaxial-to-rectangular waveguide transitions either of right-angle type or of in-line type. Figure 1 shows a commercial in-line coaxial transition in a WR34 waveguide [28]. By approximately extrapolating the manufacturer's reflection coefficient data, it can be seen that the transition operates with a reflection coefficient \(<-22.7\) dB from about 21 GHz to about 34 GHz. The start and end frequencies (21 and 34 GHz) are 1.21 and 1.96 times the fundamental TE\({}_{10}\)-mode cutoff frequency, respectively (1.62:1 bandwidth). If one can extend the bandwidth close to 3:1, it will be very useful in many applications. Especially in measurement applications, it will reduce the number of measurement accessories and the measurement time. Figure 2 shows the waveguide wall structure in the proposed design approach. The symbols \(W\), \(W_{1}\), and \(W_{2}\) denote the output waveguide, the impedance matching waveguide, and the TEM-TE\({}_{10}\) mode converter waveguide, respectively; \(a\), \(a_{1}\), and \(a_{2}\) their broad-wall width, respectively; \(b\), \(b_{1}\), and \(b_{2}\) their narrow-wall height, respectively. A stepped-ridge transformer is placed in the mode-converter waveguide \(W_{2}\). In a rectangular waveguide, the TE\({}_{mn}\)/TM\({}_{mn}\)-mode cutoff occurs at the following:  
\[f_{mn}=(c/2)\ [(m/a)^{2}+(n/b)^{2}]^{1/2} \tag{1}\]  
where \(c\) is the speed of light, \(a\) is the waveguide broad-wall width, and \(b\) is the narrow-wall height. From now on, we will use the TE modes to mean the TM modes as well since their cutoff frequencies are the same. In a standard output waveguide \(W\), where \(b\) is equal to \(a/2\), the cutoff frequency of the TE\({}_{10}\), TE\({}_{20}\), TE\({}_{11}\), and TE\({}_{30}\) modes is given by the following:  
\[f_{10}=c/(2a) \tag{2a}\] \[f_{20}=2c/(2a)=2f_{10}\] (2b) \[f_{11}=\sqrt{5}c/(2a)=2.24f_{10}\] (2c) \[f_{30}=3c/(2a)=3f_{10} \tag{2d}\]  
Since the stepped-ridge transformer is symmetric in the \(H\) plane of the mode-converter waveguide \(W_{2}\), the antisymmetric TE\({}_{20}\) mode is not excited in \(W_{2}\), and the next higher-order mode will be the TE\({}_{11}\) or TE\({}_{30}\) mode. In a standard rectangular waveguide, the TE\({}_{11}\)-mode cutoff frequency is 2.24 times the fundamental TE\({}_{10}\)-mode cutoff frequency. Thus, the bandwidth limit of a transition constructed using a standard rectangular waveguide is 2.24:1. The achieved bandwidth, however, is about 90% of the theoretical maximum or 2:1 since it is difficult to operate a transition at very close to the cutoff of the TE\({}_{10}\) and TE\({}_{11}\) modes. Figure 1: A commercial in-line coaxial-to-rectangular waveguide transition by Exceed Microwave and its reflection coefficient [28]. Figure 2: Waveguide structure in the proposed design approach (\(W\), output waveguide; \(W_{1}\), matching waveguide; \(W_{2}\), converter waveguide). At frequencies above the cutoff, the TE\({}_{30}\) mode can be generated in \(W_{2}\) along with the fundamental TE\({}_{10}\) mode and subsequently transmitted to the output waveguide \(W\). One can make a transition operate approximately from the TE\({}_{10}\) cutoff to the TE\({}_{30}\) cutoff in the output waveguide (about 3:1 bandwidth) if the TE\({}_{11}\) mode is suppressed in the waveguides \(W\), \(W_{1}\), and \(W_{2}\). If the discontinuity effect is small between \(W_{2}\) and \(W\), the generation of the TE\({}_{11}\) mode will be minimized.