# Computational Design of a Broadband In-Line Coaxial-to-Rectangular Waveguide Transition
## 3 Design Examples
An experimental approach is not suitable either because there are too many parameters to be optimized. Prior to the design of a WR75 transition, a transition with a linear taper \(T\) of zero slope (\(b=b_{1}=b_{2}\)) was designed. This aided in estimating the increased level of higher-order modes due to in the linear taper \(T\) of nonzero slope. In the final design of the WR75 transition, a linear taper with a slope was added, and dimensions were re-adjusted. Figure 5 shows the structure of a transition with a linear taper \(T\) of zero slope whose dimensions are listed in Table 1. An SMA connector was used for the coaxial input port. Figure 6 shows the reflection and transmission coefficients of the designed transition in both a decibel scale and a natural scale. The reflection coefficient is less than \(-24\) dB at 8.00-22.54 GHz (2.82:1 bandwidth). A resonance spike was generated at 21.24 GHz, causing the reflection coefficient to increase from \(-30\) dB to \(-24\) dB. Figure 4: Dimensional parameters of the proposed transition. \(b\), \(b_{1}\), and \(b_{2}\), waveguide narrow-wall height; \(L\), \(L_{1}\), \(L_{2}\), and \(L_{3}\), waveguide length; \(F\), \(D\), probe length and position; \(a\), \(a_{1}\), \(a_{2}\), waveguide broad-wall width; \(R\), radius of ridge-edge rounding; \(G\), ridge-to-back short gap; \(t\), back short thickness. Figure 5: Structure of a transition with a linear taper of zero slope: (**a**) transparent perspective view; (**b**) cross-sectional view. Figure 7 shows the transmission coefficient of the higher-order modes from the coaxial port to the output waveguide port. In the operating frequency range of 8.00-22.54 GHz, the TE\({}_{11}\) and TE\({}_{30}\) modes are predominant the highest ranging from \(-70.0\) dB to \(-40.2\) dB. The TM\({}_{11}\) mode is the next-highest one with a maximum level of \(-72\) dB at 22.54 GHz. The TE\({}_{31}\) and TM\({}_{31}\) modes are in the level of \(-130\) dB, \(-150\) dB, respectively. After confirming the performance of a transition with a linear taper of zero slope, the slope of the linear taper \(T\) was adjusted for the WR75 output waveguide, and the transition was optimized. Starting with initial dimensions, we used the \"Trust Region Framework\" optimization in CST Studio SuiteTM V2023 to obtain a final WR75 transition, whose dimensions are given in Table 2. An SMA connector was used for the coaxial input port. All the dimensional parameters except the port dimensions of the coaxial connector \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline \(a,b,L\) & 19.05, 6.57, 21.27 & \(F,D\) & 2.33, 4.59 \\ \hline \(a_{1},b_{1},L_{1}\) & 19.47, 6.57, 16.92 & \(t,G,R\) & 1.59, 0.79, 0.90 \\ \hline \(a_{2},b_{2},L_{2},L_{3}\) & 19.89, 6.57, 2.89, 23.78 & Ridge width/height & 3.55/0.91, 1.99, 3.75, 5.27 \\ \hline
\begin{tabular}{c} Coaxial \\ connector \\ \end{tabular} & \(1.27/4.11\), \(e_{7}=2.1\) (SMA) & Probe hole & Dia. = 2.95, Len. = 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the designed transition with a linear taper of zero slope. Figure 6: Reflection and transmission coefficients of a transition with a linear taper of zero slope: (**a**) in dB scale and (**b**) in natural scale. and the output WR75 waveguide were optimized. The range of the parameter values to be set in the optimization was obtained from the parametric analysis carried out prior to the optimization. Figure 8 shows the reflection coefficients during an optimization of the designed WR75 transition, where the curve in red is the reflection coefficient of the final design. In Figure 8, we observe that the reflection coefficient rapidly approaches \(-20\) dB level and that many more iterations are required to lower the reflection coefficient below \(-20\) dB. Over 200 simulations of the transition's reflection coefficient were carried out before obtaining the final design.