## 2 Circuit Description
### Non-Ideal Analysis  
The non-ideal characteristics of the CDTA are taken into account in the analysis of the proposed universal filter. The parasitic current gains between the \(p\)- and \(z\)-terminals and the \(n\)- and \(z\)-terminals, along with the non-ideal transconductance, are taken into account. Parasitic impedances are neglected, as the circuit operates at low frequencies typical for biomedical signals. In the non-ideal case, the behavior of the CDTA can be represented, as shown in [2], by the following equation:  
\[\left.\begin{array}{l}V_{p}=V_{n}=0\\ I_{z}=\alpha_{p}I_{p}-\alpha_{n}I_{n}\\ I_{x}=g_{mn}V_{z}\end{array}\right\} \tag{19}\]  
Here, \(\alpha_{p}=1-\varepsilon_{p}\), where \(\varepsilon_{p}\big{(}|\varepsilon_{p}|\ll 1\big{)}\) represents the parasitic current gain between the \(p\)- and \(z\)-terminals of the CDTA. Similarly, \(\alpha_{n}=1-\varepsilon_{n}\), where \(\varepsilon_{n}(|\varepsilon_{n}|\ll 1)\) denotes the parasitic current gain between the n- and z-terminals of the CDTA. Ideally, \(\alpha_{p}\), and \(\alpha_{n}\) are all equal to unity. The parameter \(g_{mn}\) represents the non-ideal transconductance.  
The non-ideal transconductance \(g_{mn}\) of the TA at a frequency near its cut-off can be expressed [39] as follows:  
\[g_{mn}(s)\cong g_{m}(1-sT_{o}) \tag{20}\]  
where \(T_{o}=1/\omega_{gm}\) and \(\omega_{gm}\) represents the first pole frequency of \(g_{m}\).  
Using (19) and nodal analysis, the denominator of the transfer functions can be rewritten as follows:  
\[D(s)=s^{2}C_{1}C_{2}+sC_{2}g_{m1}\alpha_{p1}+g_{m1}g_{m2}\alpha_{p1}\alpha_{n2} \tag{21}\]  
where \(\alpha_{pj}\), and \(\alpha_{nj}\) are the parasitic current gains of the \(j\)th CDTA.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filtering Function** & **Output** & **Input** & **Transfer Functions** \\ \hline \multirow{2}{*}{BSF} & \multirow{2}{*}{Non-inverting} & \(I_{o1}+I_{o6}\) & \(I_{1}\) & \(\big{(}s^{2}C_{1}C_{2}+g_{m1}g_{m2}\big{)}/D(s)\) \\ \cline{3-4}  & & \(I_{o1}+I_{o3}+I_{o5}\) & \(I_{3}\) & \(\big{(}s^{2}C_{1}C_{2}+g_{m1}g_{m2}\big{)}/D(s)\) \\ \hline APF & Non-inverting & \(I_{o1}+I_{o5}+I_{o6}\) & \(I_{1}\) & \(\big{(}s^{2}C_{1}C_{2}-sC_{2}g_{m1}+g_{m1}g_{m2}\big{)}/D(s)\) \\ \hline \hline \end{tabular} Note: \(D(s)=s^{2}C_{1}C_{2}+sC_{2}g_{m1}+g_{m1}g_{m2}\).  
\end{table}
Table 1: Cont.  
Considering the non-idealities in (20) for \(g_{m1}\) and \(g_{m2}\), the non-idealities can be approximated as \(g_{m1}(s)\cong g_{m1}(1-sT_{01})\) and \(g_{m2}(s)\cong g_{m2}(1-sT_{02})\). The denominator \(D(s)\) of the transfer functions of the filters can be then written as follows:  
\[\begin{array}{c}D(s)=s^{2}C_{1}C_{2}\Big{(}1-\frac{C_{2}g_{m1}g_{1}T_{c1}-g_{ m1}g_{m2}g_{1}g_{m2}g_{1}g_{m2}g_{1}g_{m2}g_{1}g_{m2}g_{1}g_{m2}}\Big{)}\\ \hskip 14.226378 pt+sC_{2}g_{m1}g_{1}\Big{(}1-\frac{g_{m1}g_{m2}g_{m2}(g_{1}+g_{ m2})}{C_{2}g_{m1}}\Big{)}+g_{m1}g_{m2}\alpha_{p1}\alpha_{n2}\end{array} \tag{22}\]  
From (22), the non-ideal effects of the transconductances \(g_{m1}\) and \(g_{m2}\) can be mitigated by satisfying the following conditions:  
\[\frac{C_{2}g_{m1}g_{1}T_{c1}-g_{m1}g_{m2}g_{1}\alpha_{n2}T_{c1}T_{c2}}{C_{1}C_ {2}}\ll 1 \tag{23}\]  
\[\frac{g_{m1}g_{m2}\alpha_{n2}(T_{c1}+T_{c2})}{C_{2}g_{m1}}\ll 1 \tag{24}\]  
These conditions can be achieved by appropriately selecting the values of \(g_{m1}\) and \(g_{m2}\) and capacitances \(C_{1}\) and \(C_{2}\).  
In the non-ideal case, the natural frequency and quality factor of the proposed filter become  
\[\omega_{o}=\sqrt{\frac{g_{m1}g_{m2}\alpha_{p1}\alpha_{n2}}{C_{1}C_{2}}} \tag{25}\]  
\[Q=\sqrt{\frac{g_{m2}C_{1}\alpha_{n2}}{g_{m1}C_{2}\alpha_{p1}}} \tag{26}\]