## 3 Modified and Simplified T-Type Structure Designed for Low Attenuation Units  
### Analysis of the Mechanism Contributing to the Phase Error  
Among the various metrics of the DSAs, amplitude/phase characteristics, IL and voltage standing wave ratio have received the most attention. Although T-type and \(\pi\)-type structures can provide favorable control of attenuation accuracy with proper impedance matching, there is no avoidance of the extra path loss introduced by the transistor on-state resistance \(R_{on}\). In order to achieve a high-precision attenuation cell with low IL in a compact layout, we applied a simplified T-type cell, whose circuit model considering the parasitic parameters is illustrated in Figure 3a. To clarify the mechanisms jeopardizing the phase error of the unit branch, we start from the analysis of the proposed attenuator operated in reference mode and attenuation mode, respectively. Figure 2: Cross-sectional view of the adopted GaAs technology. Figure 3b,c shows the equivalent circuits for the reference and attenuation states, respectively. As immediately seen, when turned off (i.e., reference mode), the transistor can be treated as a series connection of resistance \(R_{s}\) (i.e., \(R_{\textit{off}}\)) and capacitance \(C_{s}\) (i.e., \(C_{\textit{off}}\)), which are characterized by the transistor's parasitic parameters as  
\[R_{s}=\frac{R_{p}\left(C_{\textit{gd}}+C_{\textit{gs}}\right)^{2}}{\omega^{2}{ R_{p}}^{2}\left(C_{\textit{gd}}C_{\textit{gs}}+C_{\textit{gd}}C_{\textit{ds}}+C_{ \textit{gs}}C_{\textit{ds}}\right)^{2}+\left(C_{\textit{gd}}+C_{\textit{gs}} \right)^{2}} \tag{1}\]  
\[C_{s}=\frac{C_{\textit{gd}}C_{\textit{gs}}+C_{\textit{gd}}C_{\textit{ds}}+C_{ \textit{gs}}C_{\textit{ds}}}{C_{\textit{gd}}+C_{\textit{gs}}}+\frac{C_{\textit {gd}}+C_{\textit{gs}}}{\omega^{2}{R_{p}}^{2}\left(C_{\textit{gd}}C_{\textit{gs} }+C_{\textit{gd}}C_{\textit{ds}}+C_{\textit{gs}}C_{\textit{ds}}\right)} \tag{2}\]  
where \(\omega\) is the angular frequency of interest, and \(R_{p}\), \(C_{\textit{gs}}\), \(C_{\textit{gd}}\) and \(C_{\textit{ds}}\) represent the parasitic resistance, gate-to-source parasitic capacitance, gate-to-drain parasitic capacitance and drain-to-source parasitic capacitance of the transistor, respectively. In this case, since the \(C_{\textit{off}}\) is extremely small, it is the dominant component contributing to isolation between the RF signal to ground. As a result, the phase characteristics exhibit a low-pass effect, which can be calculated as the following equation:  
\[\theta_{\textit{ref1}}=-\tan^{-1}\left(\frac{\omega C_{\textit{off}}Z_{0}}{2+ \omega^{2}C_{\textit{off}}^{2}R_{\textit{off},\textit{all}}\left(2R_{\textit{ off},\textit{all}}+Z_{0}\right)}\right) \tag{3}\]  
where \(Z_{0}\) denotes the characteristic impedance and \(R_{\textit{off},\textit{all}}=R_{\textit{off}}+R_{1}\). On the other hand, when the transistor is turned on (i.e., attenuation mode), it can be treated as a parallel connection of resistance \(R_{p}\) (i.e., \(R_{\textit{on}}\)) and capacitance \(C_{p}\) according to the equivalence principle, which is derived as  
\[C_{p}=\frac{C_{\textit{gd}}C_{\textit{gs}}+C_{\textit{gd}}C_{\textit{ds}}+C_ {\textit{gs}}C_{\textit{ds}}}{C_{\textit{gd}}+C_{\textit{gs}}} \tag{4}\]  
At this point, the impedance of \(C_{p}\) is considerably larger to negligible compared to \(R_{p}\), thereby the parasitic inductance \(L_{s}\) (i.e., \(L_{\textit{on}}\), which contains metal lines connected to the source and drain of the transistor) and \(R_{p}\) play a major factor, causing a high-pass effect. Its equivalent phase in the attenuation mode is calculated as  
\[\theta_{\textit{att1}}=\tan^{-1}\left(\frac{\omega L_{\textit{on}}Z_{0}(R_{ \textit{on},\textit{all}}+Z_{0})}{2\omega^{2}L_{\textit{on}}^{2}Z_{0}+Z_{0}R_{ \textit{on},\textit{all}}\left(R_{\textit{on},\textit{all}}+\frac{1}{2}Z_{0} \right)}\right) \tag{5}\]  
where \(R_{\textit{on},\textit{all}}=R_{\textit{on}}+R_{1}\).