# Design of a Broadband Transition from a Coaxial Cable to a Reduced-Height Rectangular Waveguide
## 3 Design
For these waveguide dimensions, the TE\({}_{10}\) mode cutoff is at 7.87 GHz, and the TE\({}_{11}\)/TM\({}_{11}\) mode cutoff in the output waveguide occurs at 63.45, 32.44, and 22.41 GHz for \(b=0.125a\), \(0.25a\), and \(0.375a\), respectively. The parameter \(L\) is the output WR75 waveguide length whose value does not affect the transition performance as long as it is large enough. A value of 15 mm or about 0.787\(a\) was used in all cases. The width \(a_{1}\) of the first waveguide is set about 4% larger than the output waveguide width \(a\) so that its TE\({}_{10}\) mode is cut off at 7.57 GHz. The height \(b_{1}\) of the first waveguide is set at a value 8% smaller than that of the output waveguide height \(b\). In this case, the TE\({}_{11}\)/TM\({}_{11}\) mode cutoff in the first waveguide occurs at 68.87, 35.10, and 24.13 GHz for \(b=0.125a\), \(0.25a\), and \(0.375a\), respectively. The width \(a_{4}\) and height \(h_{4}\) of the second waveguide are initially set at the average of those of the other two waveguides. The length \(L_{4}\) of the second waveguide is set at 0.3 times the output waveguide width \(a\). One of the most critical dimensions is the probe location \(L_{1}\), whose initial value is set at \(0.25a\). The disk diameter \(D_{1}\) and height \(H_{1}\) are initially set at \(0.10a\) and \(0.50b\) and adjusted for a wideband operation. The dual tuning posts' location \(L_{2}\) and \(S_{2}\), diameter \(D_{2}\), and height \(H_{2}\) are adjusted to reduce the reflection coefficient to a value less than \(-20\) dB with their initial values set at \(0.25a\), \(0.35a\), \(0.20a\), and \(0.25b\), respectively. In Table 1, we provided the initial values of the dimensional parameters normalized with the output waveguide width \(a\) or with the height \(b\), depending on the parameter type. For the coaxial cable, we used an SMA connector with an outer conductor diameter  
Figure 2: Detailed structure and dimensional parameters of the proposed transition. \(2b_{0}=4.11\) mm and an inner conductor diameter of \(2a_{0}=1.27\) mm. PTFE material (\(\epsilon_{r}=2.08\)) fills the space between the inner and outer conductors of the coaxial line. The first higher-order TE\({}_{11}\) mode cutoff occurs at 24.58 GHz in this coaxial line. Figures 3 and 4 show changes in the reflection coefficient of the transition due to variations in the disk probe and tuning post dimensions of the transition with \(b/a=0.125\). These figures can also serve as tolerance analysis results. We observe that the reflection coefficient is sensitive to disk height \(  
After extensive parametric analyses, we set the parameters for the automatic optimization by the simulation tool: the range of dimensional parameter values, the operating frequency range, and the target reflection coefficient value. Out of the seven optimization techniques in the CST Studio Suite, such as 'Genetic Algorithm' and 'Particle Swarm Optimization', we found that the 'Trust Region Framework' was the most efficient and reliable for this problem. The number of simulations with the 'Trust Region Framework' optimization ranges roughly from 50 to 300. Figure 5 shows the variation in the reflection coefficient during the optimization of the transition with \(b/a=0.125\). Figure 6 shows the electric and magnetic fields inside the transition with \(b/a=0.375\). Operation with well-matched impedance can be observed by noting that the maximum field is nearly uniform along the wave propagation path, resulting from the small value for the voltage standing wave ratio. Table 2 shows the dimensions of the optimized transitions with \(b/a\) ratios of 0.125, 0.250, and 0.375.