# Computational Design of a Broadband In-Line Coaxial-to-Rectangular Waveguide Transition
## 3 Design Examples  
Figure 3 shows the structure of an in-line coaxial-to-rectangular waveguide transition to be designed according the approach proposed in this Communication. The output waveguide \(W\) is a standard waveguide with broad-wall width \(a\) and narrow-wall height \(b\) (\(b=a/2\) in most of the standard waveguides). The converter waveguide \(W_{2}\) has broad-wall width \(a_{2}\) slightly larger than \(a\) (as given by Equation (6)) and narrow-wall height \(b_{2}\) smaller than \(b\) (as given by Equation (5)). The matching waveguide \(W_{1}\) is used to ensure a broadband impedance matching between the converter waveguide \(W_{2}\) and output waveguides \(W\), which have different broad-wall widths. A linear taper \(T\) in the narrow-wall height is used in the waveguides \(W_{1}\) and \(W\) for design simplicity and for a smooth change in the narrow-wall height from \(b_{2}\) to \(b\). One may use a stepped waveguide transformer, in which case there will be more parameters to be optimized. If necessary, the linear taper may have a zero slope or a negative slope to make the height of the output waveguide \(W_{1}\) less than or equal to that of the converter waveguide \(W_{2}\). A stepped-ridge transformer \(S\) converts the TEM mode of the coaxial connector \(X\) to the TE\({}_{10}\) mode of the converter waveguide \(W_{2}\). The coaxial probe \(P\) is connected to the stepped-ridge transformer \(S\) across a cavity \(C\) through a probe hole \(H\) in the back short \(B\). The four-step ridge transformer \(S\) plays an important role, as it is crucial in determining the level of the reflection coefficient and the degree of the higher-order mode generation. A stepped-ridge transformer seems to be better in achieving a low reflection coefficient over a broad bandwidth than a continuously tapered transformer since it has more control parameters. The number of steps in the ridge transformer is important for broadband operation. It has been found that four steps are sufficient for our transition design. The edges of the first ridge--the one connected to the coaxial probe \(P\)--are rounded in the circular shape of radius \(R\) for better impedance matching. In addition to \(S\), the cavity \(C\) and the probe hole \(H\) play a role in achieving a low reflection coefficient over a broad frequency range. Figure 4 shows the dimensional parameters of the transition. With the given dimensions \(a\) and \(b\) of the output waveguide \(W\), the design of the transition starts with determining \(a_{2}\) and \(b_{2}\) of the converter waveguide \(W_{2}\), which was explained in the previous section. The height \(b_{2}\) of the converter waveguide \(W_{2}\) is held constant throughout its length to suppress the generation of the TE\({}_{11}\) mode. The length (\(L_{2}+L_{3}\) in Figure 4) of the converter waveguide \(W_{2}\) is one of the optimization parameters. Figure 3: Structure of the proposed transition. \(W\), output waveguide; \(W_{1}\), matching waveguide; \(W_{2}\), mode-converter waveguide; \(S\), stepped-ridge transformer; \(C\), matching cavity; \(B\), back short; \(P\), coaxial probe; \(X\), coaxial connector; \(H\), probe hole; \(T\), linear taper. The width \(a_{1}\) and the length \(L_{1}\) of the matching waveguide are among the parameters to be optimized. The length of a linear taper \(T\) is the sum of the output waveguide length \(L\) and the matching waveguide length \(L_{1}\), which is optimized for a reflection coefficient \(<-20\) dB, especially at the start frequency in the operating bandwidth. The dimensions of the four-step ridge transformer \(S\), the cavity \(C\), and the probe hole \(H\) can be optimized by using a powerful modern simulation tool such as CST Studio SuiteTM V2023. The ridge-edge-rounding radius \(R\) is held at a constant value. It is virtually impossible to obtain a low reflection coefficient over a broad frequency range by an analytical method since the calculation of the reflection coefficient is extremely complicated if not impossible.