## 4 High Attenuation Units and Level Shifter Structure Design  
Analysis and Design of Modified \(\pi\)-Type and Switched-Path Type Structures for 4 dB, 8 dB and 16 dB Units  
From the discussion in Section 3.3, it is clear that the proposed structure was no longer suitable for high attenuation cells. Instead, the \(\pi\)-type topology shown in Figure 9a was  
Figure 8: Comparative simulation results with traditional bridge T-type 2 dB attenuation structure: (a) \(S_{21}\); (b) \(S_{11}\); and (c) phase error. Figure 7: Simulation results for the proposed 0.5 dB, 1 dB and 2 dB attenuation structures: (a) relative attenuation; (b) \(S_{11}\); and (c) phase error. adopted as it has a larger bandwidth after compensation compared to the T-type topology. According to the reference and attenuation state equivalent circuits in Figure 9b,c, the transfer functions for their respective states can be derived as follows:  
\[T(s)_{\pi,ref}=\frac{2Z_{0}\Big{[}1+C_{off2}R_{p}s\Big{]}^{2}}{(a_{1}s+1)(a_{2}s +b_{2})} \tag{10}\]  
\[T(s)_{\pi,att}=\frac{2Z_{0}\Big{(}1+C_{off1}R_{s}s\Big{)}\big{(}R_{p}+R_{on2} \big{)}^{2}}{\big{(}R_{p}+R_{on2}+Z_{0}\big{)}(a_{3}s+b_{3})} \tag{11}\]  
where \(a_{1}=C_{off2}(R_{p}+Z_{0})\), \(a_{2}=C_{off2}(c_{2}R_{p}+Z_{0}R_{B})\), \(b_{2}=2Z_{0}+R_{B}\), \(R_{B}=\)\(R_{s}\)\(\mid\)\(R_{on1}\), \(a_{3}=2C_{off1}R_{s}Z_{0}(R_{p}+R_{on2})\), and \(b_{3}=R_{s}(R_{p}+R_{on2}+Z_{0})+2\)\(Z_{0}(R_{p}+R_{on2})\). It is well known that when zeros and poles occur at the same or a closely approximated frequency, they cancel out each other and thus produce a flat response. As can be learned from Equations (10) and (11), it is rather difficult to satisfy pole-zero cancelation relying solely on resistor selection and transistor sizing. However, when additional compensating components are introduced, more poles and zeros are tacked on to their numerator and denominator, resulting in more opportunities for tuning. Here, we applied the compensation scheme of the series inductor, as shown in Figure 9d. By analyzing the updated derived transfer function, it is able to find the critical zeros and poles to obtain a flat amplitude/phase characteristic. In order to visualize the effect and avoid complicated formulas, the simulated performance of the 4 dB unit with various \(L_{T}\) is given in Figure 10. In view of the results, an appropriate increase in \(L_{T}\) favored the bandwidth expansion of the relative attenuation, while an optimal compensation value existed for the phase error. Therefore, the value of \(L_{T}\) needs to be considered as a tradeoff according to the actual situation. The initial values of the parameters adopted for the 4 dB and 8 dB units are shown in Table 1, which are relatively idealized components. Immediately, they should be further iteratively optimized based on the co-simulation performance after cascade design. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(R_{s}\) & \(R_{p}\) & \(L_{1}\) & \(L_{T}\) & \(M_{1}\) & \(M_{2}\) \\ \hline
4-dB & 21.3 \(\Omega\) & 109 \(\Omega\) & 86.5 pH & 0.315 nH & 4 \(\times\) 50 \(\upmu\)m & 1 \(\times\) 15 \(\upmu\)m \\ \hline
8-dB & 35.3 \(\Omega\) & 13.5 \(\Omega\) & 67.5 pH & N/A & 4 \(\times\) 25 \(\upmu\)m & 1 \(\times\) 15 \(\upmu\)m \\ \hline \hline \end{tabular} N/A Not applicable.  
\end{table} 
Table 1: Parameter values for 4 dB and 8 dB units. Figure 9: Schematics of (**a**) typical \(\uppi\)-type structure1; (**b**) reference-state equivalent circuit; (**c**) attenuation-state equivalent circuit; and (**d**) modified \(\uppi\)-type with series inductor. As for the 16 dB unit, the attenuation is so large that either the \(\pi\)-type or T-type is no longer valid, so we employed the switched-path type in Figure 1c. The proposed performances of the 4 dB, 8 dB and 16 dB units are shown in Figure 11, exhibiting good accuracy in attenuation and phase error.