# Design of a Broadband Transition from a Coaxial Cable to a Reduced-Height Rectangular Waveguide
## 3 Design  
In this communication, we present a simulation-based design method for a broadband CWT for reduced-height rectangular waveguides. This work is not an outcome of a research project which would warrant the fabrication and measurement of a prototype design. As such, the validation of the proposed concept was conducted through a computer simulation using CST Studio Suite, a well-known and accurate tool for EM simulation. We provide extensive details on the design of the proposed CWT so that the achieved results can be verified by anyone if needed. Designs are presented for a coaxial-to-rectangular waveguide transition with a waveguide height(\(b\))-to-width(\(a\)) ratio \(b/a\) ranging from 0.125 to 0.375 having a ratio bandwidth of \(>\)2.4 or, equivalently, a percent bandwidth of \(>\)82.4%. A transition's bandwidth is determined from the frequency range\(f_{1}\leq f\leq f_{2}\), where the reflection coefficient is less than, for example, \(-\)20 dB, or the VSWR is less than 1.2. Then the ratio bandwidth is given by\(f_{2}/f_{1}\) and the percent bandwidth by \(200(f_{2}-f_{1})/(f_{2}+f_{1})\). Figure 1 shows the structure of the proposed transition together with a rectangular coordinate system whose origin is at the lower left corner in the back short of the waveguide \(R_{1}\). Waveguide walls are assumed to be made of a perfect electric conductor, and the waveguide interior is filled with air. To implement the proposed CWT, we considered employing a tuning post in addition to a coaxial probe, as reported by [PERSON] and coworkers [4]. They analyzed and designed a coaxial-to-waveguide adaptor with a descended probe and a tuning post. The tuning post served as a short-circuited second probe and was used to tune the impedance matching. When the coaxial probe was positioned at the waveguide's half-width position \(x=a/2\), the TE\({}_{20}\) mode was suppressed. Similar results were obtained with two tuning posts symmetrically placed in the \(x\) direction. Here, we leverage this technique to increase the bandwidth. Figure 2 shows the detailed structure and dimensional parameters of the proposed transition. For a broader frequency range of operation, the starting frequency should be lowered and the upper frequency increased. To lower the starting frequency, we leveraged the Cano-Mediavilla technique for increased bandwidth, which achieved a bandwidth ratio of 2.0 [3]. In this method, a double-step conversion of the rectangular waveguide width is employed. This allows the transition to operate close to the TE\({}_{10}\) mode cutoff of the output waveguide. In designing a wideband rectangular waveguide transition, it is important to check the cutoff frequencies of the higher-order modes as well as that of  
Figure 1: Structure of the proposed coaxial-to-rectangular waveguide transition: (**a**) transparent perspective view; (**b**) cutaway view: \(R\), output waveguide; \(R_{4}\), second waveguide; \(R_{1}\), first waveguide; \(S\), tuning post; \(D\), disk; \(P\), probe; C, coaxial cable; \(B\), back short. the dominant TE\({}_{10}\) mode. The cutoff frequency \(f_{c}\) and the guided wavelength \(\lambda_{g}\) of the TE\({}_{mn}\)/TM\({}_{mn}\) mode in an air-filled rectangular waveguide is given by  
\[f_{c}=(c/2)\ [(m/a)^{2}+(n/b)^{2}]^{1/2} \tag{1}\]  
\[\lambda_{g}=\lambda/[1-(f_{c}/f)^{2}]^{1/2} \tag{2}\]  
where \(c\) is the speed of light in vacuum, \(a\) is the broad-wall width, \(b\) is the narrow-wall height, and \(\lambda\) is the wavelength in vacuum. Starting from the initial dimensions, we carried out a parametric analysis of the structure prior to the automatic optimization using CST Studio Suite. The output waveguide width \(a\) is the same as that of the WR75 standard waveguide (19.05 mm), and the output waveguide height \(b\) is \(0.125a\), \(0.25a\), and \(0.375a\).