## 3 Limitations of Conventional Structures  
In general, attenuation bits with large attenuation values contribute large amplitude and phase variations and limit the bandwidth. In this section, the limitations of the conventional structure are analyzed using a conventional \(\pi\)-type attenuator as an example.  
Figure 4 shows the equivalent circuit of the conventional \(\pi\)-type attenuator. Considering the parasitic capacitance of the transistor, the equivalent circuit of the reference state can be considered as a low-pass filter with phase lag, and the parasitic capacitance _Cp,off_ of the shunt transistor will cause the leakage of the high-frequency signal to ground. Conversely, the equivalent circuit of the attenuation state can be considered as a high-pass filter with phase advance, and the high-frequency signal can be directly output through the parasitic capacitance _Cs,off_ of the series transistor. Consequently, the presence of _Cs,off_ and _Cp,off_ leads to a relative attenuation value that is less than the ideal one, accompanied by an additional phase shift exceeding \(0^{\circ}\).  
The transmission matrix for a conventional \(\pi\)-type structure in the reference state is given by the following (13):  
\[\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}1+\frac{\omega^{2}C_{poff}{}^{2}R_{p}R_{s,on}R_ {S}}{R_{s}+R_{s,on}}+\frac{\omega C_{poff}R_{s,on}R_{s}}{R_{s}+R_{s,on}}&\frac {R_{s}R_{s,on}}{R_{s}+R_{s,on}}\\ 2\omega^{2}C_{poff}{}^{2}(R_{p}-\frac{R_{s}R_{s,on}}{R_{s}+R_{s,on}})+j2\omega C _{poff}&1+\fracThe corresponding transmission S-parameter is given by the following (14):  
\[S_{21}=(\frac{2}{A+\frac{B}{Z_{0}}+Z_{0}C+D})=\frac{1}{1+\frac{R_{s,on}R_{s}}{K_{ s,on}+R_{s}}(\omega^{2}C_{p,aff})^{2}R_{P}+j\omega C_{p,off}+\frac{1}{2Z_{0}})+ \omega Z_{0}C_{p,off}(\omega C_{p,off}R_{p}-\frac{\omega C_{p,off}R_{s}}{K_{ s,on}+R_{s}}+j) \tag{14}\]  
where _Cp,off_ is the parasitic capacitance in the off state of the parallel transistor.  
The transmission phase in the reference state can be expressed as follows:  
\[\varphi_{R}\approx-\tan^{-1}\frac{2\omega Z_{0}^{2}C_{p,off}(R_{s,on}+R_{s})^ {2}(R_{s,on}R_{s}+R_{s,on}Z_{0}^{2}+R_{s}Z_{0}^{2})}{R_{s,on}R_{s}} \tag{15}\]  
Similarly, the transmission phase in the attenuation state can be expressed as follows:  
\[\varphi_{A}\approx-\tan^{-1}\frac{\omega C_{s,off}(Z_{0}^{2}+R_{p,on}R_{p})}{2 R_{p,on}Z_{0}^{2}+2R_{p}Z_{0}^{2}} \tag{16}\]  
where \(\omega^{2}C_{s,off}{}^{2}R_{s}{}^{2}\) and \(\omega^{2}C_{p,off}{}^{2}R_{p}{}^{2}\) are omitted to simplify the calculations.  
In order for the transmission phase difference to be zero, the following equation must be satisfied:  
\[\Delta\varphi=\varphi_{A}-\varphi_{R}=0 \tag{17}\]  
The transmission phase \(\varphi_{R}\) is less than or equal to 0, while \(\varphi_{A}\) is greater than or equal to 0. Therefore, to achieve zero transmission phase difference, the following relationship should be satisfied:  
\[\varphi_{A}=\varphi_{R}=0 \tag{18}\]  
However, the parasitic capacitance of the transistor cannot be zero, so the phase of the attenuation state of the conventional \(\pi\)-type topology is always ahead of the phase of the reference state. Additionally, as the frequency increases, the reactance of the capacitor decreases, which can exacerbate the amplitude and phase errors between the two states.