## 2 Circuit Descriptions
### Non-Ideal Analysis and Sensitivity Performance  
If the non-idealities of CCCCTA are considered, the relationships of the terminal voltages and currents can be rewritten as V\({}_{\text{X}}=\beta\)V\({}_{\text{Y}}+\)I\({}_{\text{X}}\)R\({}_{\text{X}}\), I\({}_{\text{Z}+}=\alpha_{\text{p}}\)I\({}_{\text{X}}\), I\({}_{\text{Z}-}=-\alpha_{\text{n}}\)I\({}_{\text{X}}\), I\({}_{\text{O}1}=\gamma_{\text{p}}\)g\({}_{\text{m}}\)V\({}_{\text{Z}+}\), I\({}_{\text{-O}1}=-\gamma_{\text{n}}\)g\({}_{\text{m}}\)V\({}_{\text{Z}+}\), \(\text{I}_{\text{O}2}=\eta_{\text{p}}\)g\({}_{\text{m}}\)V\({}_{\text{Z}+}\) and I\({}_{\text{-O}2}=-\eta_{\text{n}}\)g\({}_{\text{m}}\)V\({}_{\text{Z}+}\), where \(\beta\), \(\alpha_{\text{p}}\), \(\alpha_{\text{n}}\), \(\gamma_{\text{p}}\), \(\gamma_{\text{n}}\), \(\eta_{\text{p}}\) and \(\eta_{\text{n}}\) are CCCCTA transfer ratios that deviate from unity by the transfer errors [27]. In a non-ideal case with reanalysis of the proposed circuit in Figure 2, the denominator of the non-ideal voltage transfer function is yielded as follows:  
\[\text{D}(\text{s})=\beta_{1}\text{s}^{2}\text{C}_{1}\text{C}_{2}\text{G}_{ \text{X}1}+\alpha_{\text{p}1}\beta_{1}\gamma_{\text{p}1}\text{S}\text{C}_{1} \text{g}_{\text{m}1}\text{G}_{\text{X}1}+\alpha_{\text{p}1}\alpha_{\text{p}2} \beta_{1}\beta_{2}\gamma_{\text{p}2}\text{g}_{\text{m}2}\text{G}_{\text{X}1} \text{G}_{\text{X}2} \tag{35}\]  
where \(\beta_{\text{i}}\), \(\alpha_{\text{pi}}\) and \(\gamma_{\text{pi}}\) are the parameters \(\beta\), \(\alpha_{\text{p}}\) and \(\gamma_{\text{p}}\), respectively, of the ith CCCCTA (i = 1, 2, 3).  
The non-ideal expressions for \(\omega_{\text{o}}\) and Q are obtained as follows:  
\[\omega_{\text{o}}=\sqrt{\frac{\alpha_{\text{p}1}\alpha_{\text{p}2}\beta_{2} \gamma_{\text{p}2}\text{g}_{\text{m}2}\text{G}_{\text{X}2}}{\text{C}_{1}\text {C}_{2}}},\,Q=\frac{1}{\gamma_{\text{p}1}\text{g}_{\text{m}1}}\sqrt{\frac{ \alpha_{\text{p}2}\beta_{2}\gamma_{\text{p}2}\text{g}_{\text{m}2}\text{G}_{ \text{X}2}\text{C}_{2}}{\alpha_{\text{p}1}\text{C}_{1}}} \tag{36}\]  
The active and passive sensitivities of the proposed circuit are  
\[\text{S}_{\text{x}\text{p}1}^{\omega_{\text{o}}}=\text{S}_{\alpha_{\text{p}2} }^{\omega_{\text{o}}}=\text{S}_{\beta_{2}}^{\omega_{\text{o}}}=\text{S}_{ \gamma_{\text{p}2}}^{\omega_{\text{o}}}=\text{S}_{\text{g}_{\text{m}2}}^{ \omega_{\text{o}}}=\text{S}_{\text{G}_{\text{x}\text{O}2}}^{\omega_{\text{o}}}= -\text{S}_{\text{C}_{1}}^{\omega_{\text{o}}}=-\text{S}_{\text{C}_{2}}^{\omega_ {\text{o}}}=\frac{1}{2} \tag{37}\]  
\[\text{S}_{\text{x}\text{p}2}^{\text{Q}}=\text{S}_{\beta_{2}}^{\text{Q}}=\text{S }_{\gamma_{\text{p}2}}^{\text{Q}}=\text{S}_{\text{g}_{\text{m}2}}^{\text{Q}}= \text{S}_{\text{G}_{\text{x}\text{O}2}}^{\text{Q}}=\text{S}_{\text{C}_{2}}^{ \text{Q}}=-\text{S}_{\text{C}_{1}}^{\text{Q}}=\frac{1}{2} \tag{38}\]  
\[\text{S}_{\gamma_{\text{p}1}}^{\text{Q}}=\text{S}_{\text{g}_{\text{m}1}}^{\text{Q}}=-1 \tag{39}\]  
These calculation results indicate that all sensitivities are low and have absolute values no larger than unity. The proposed circuit thus exhibited low sensitivity.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Filter function**} & \multicolumn{2}{c}{**V\({}_{\text{in}}\) = 0, I\({}_{\text{in}}\) is Input Signal**} & \multicolumn{2}{c}{**I\({}_{\text{in}}\) = 0, V\({}_{\text{in}}\) is Input Signal**} \\ \cline{2-5}  & **CM** & **TIM** & **VM** & **TAM** \\ \hline HP & I\({}_{01}\) & V\({}_{03}\) & V\({}_{03}\) & I\({}_{01}\) \\ LP & I\({}_{03}\) & V\({}_{01}\) & V\({}_{01}\) & I\({}_{03}\) \\ BP & I\({}_{02}\), I\({}_{04}\), I\({}_{05}\) & V\({}_{02}\), V\({}_{04}\) & V\({}_{02}\) & I\(