# A Back-to-Back Gap Waveguide-Based Packaging Structure for E-Band Radio Frequency Front-End
## 2 Unit Design of the Packaging Structure
### MS-GW Transition and Connecting Structure Design  
The proposed transition from a MS to a GW in the amplifier application is shown in Figure 11. Within the GW, the signal undergoes conversion through multiple metal ridges of varying heights and is then tightly coupled to the MS line using a rectangular patch structure with dimensions matching those of \(w\)r and \(l\)m1 at the end of each ridge, facilitating energy transfer to the MS line. Then, the impedance is gradually transferred to a 50 \(\Omega\) MS line with a width of \(w\)1 through a trapezoidal structure of length \(l\)m1. The dielectric plate material of the MS line is quartz with a thickness of 0.1 mm (\(l\)q = 0.1 mm). Furthermore, periodic metal square columns are introduced at the junction between each ridge end and the MS line, spaced apart by \(w\)q, serving as suppressors for waveguide modes. The MS line matching is designed based on the theory of the gradient line matching theory, while the ridge waveguide matching process is designed using a fourth-order Chebyshev converter. The impedance of each order can be calculated according to the impedance converter theory. Typically, the length of the ridge waveguide chosen in the Chebyshev converter is one-quarter of the wavelength. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameters** & \(w\) & \(l\)**1** & \(l\)**2** & \(l\)**3** & \(l\)**t** & \(l\)**ii** & \(l\)**ii** & \(l\)**1** & \(h\)**2** \\ \hline \multirow{2}{*}{value} & BPF\_H & 2.8 & 2.225 & 2.235 & 2.226 & 0.1 & 0.8 & 0.45 & 0.563 & 0.646 \\ \cline{2-11}  & BPF\_L & 3.1 & 2.759 & 2.746 & 2.724 & 0.1 & 0.8 & 0.42 & 0.502 & 0.549 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the BPFs (Unit: mm). Figure 11: Structure and dimensions of the MS–GW transition. Figure 10: Simulated results for the BPFs. Based on the Chebyshev matching theory, the maximum reflection coefficient of the matching unit can be calculated as  
\[A=\ln(\frac{Z_{L}}{Z_{GGW}})\cdot\frac{1}{2T_{N}(\sec\theta_{M})} \tag{5}\]  
where \(Z_{L}\) is the load impedance (standard waveguide), and \(Z_{GGW}\) is the input impedance (gap waveguide). And the \(\theta_{M}\) can be calculated by  
\[\theta_{M}=\frac{\pi}{2}(1-\frac{\Delta f}{2f_{0}}) \tag{6}\]  
where \(\Delta f\) and\(f_{0}\) are the bandwidth and center frequency of the transition structure, respectively. The impedance and the impedance of the gap waveguide sections can be computed by [24]  
\[Z_{\mathrm{gw}}=2\frac{\eta_{0}}{\sqrt{\varepsilon_{r}}}\frac{b}{w} \tag{7}\]  
where \(\eta_{0}\) is the vacuum impedance, \(\varepsilon_{r}\) is the dielectric constant of the GW filled medium, and \(b\) and \(w\) are the height and width of the GW. According to the formulas presented above, the initial values of the transition dimensions can be determined, and the final dimensions can subsequently be obtained through optimization. Due to the implementation of this structure in the amplifier packaging, a back-to-back structure design is necessary. The simulated results of the final configuration are depicted in Table 2 and Figure 12, within the frequency band range of 70-88 GHz; the return loss surpasses 25 dB, while the insertion loss remains below 0.5 dB, demonstrating exceptionally low transition loss. In addition, it is crucial to employ a 90\({}^{\circ}\) angle structure for connection in order to complete the final packaging structure design and minimize the overall volume. As \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters** & **_I_m1** & **_I_m2** & **_w_m1** & **_w_r_** & **_w_q** & **_I_q** & **_I_r_1** \\ \hline Value & 0.385 & 0.825 & 0.22 & 1 & 1.8 & 10 & 0.113 \\ \hline
**Parameters** & **_hr_2** & **_hr_3** & **_hr_4** & **_lr_1** & **_lr_2** & **_lr_3** & **_lr_4** \\ \hline Value & 0.359 & 0.607 & 0.742 & 1.374 & 1.141 & 0.843 & 1.175 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimensions of the transition. (Unit: mm). Figure 12: Simulated results of the MS-GW back-to-back transition.