# Single EX-CCCII-Based First-Order Versatile Active Filter
## 2 Circuit Description
### Applications to a Mixed-Mode First-Order Active Filter  
Thanks to its low-input and high-output current-mode, the proposed current-mode active filter in Figure 3 can be applied to work as a TIM, TAM, and VM filter. Figure 4a shows the proposed TIM filter. From the transfer functions (3)-(6) of the current-mode filter, and using resistors \(R_{out}\) to convert the current outputs \(I_{01}\), \(I_{02}\), \(I_{03}\), and \(I_{04}\) to the voltage outputs \(V_{01}\), \(V_{02}\), \(V_{03}\), \(V_{04}\), \(V_{AP+}\), and \(V_{AP-}\), the output voltages of the TIM filter can be expressed by  
\[\text{HPF}\qquad\quad V_{01}=-V_{02}=R_{out}\bigg{(}\frac{sC_{1}R_{x1}}{sC_{1} R_{x1}+1}\bigg{)}\,I_{in} \tag{8}\]  
\[\text{LPF}\qquad\quad V_{03}=-V_{04}=R_{out}\bigg{(}\frac{1}{sC_{1}R_{x1}+1} \bigg{)}\,I_{in} \tag{9}\]  
\[\text{APF}+\qquad\quad\quad\quad\quad  
Figure 4b shows the proposed TAM filter. In this case, the resistor \(R_{in}\) is used to convert the voltage input \(V_{in}\) to the current input \(I_{in}\). From the transfer functions (3)-(6) of the current-mode filter, the output currents of the TAM filter can be expressed by  
\[\text{HPF}\qquad\quad I_{o2}=-I_{o1}=\frac{1}{R_{in}}\bigg{(}\frac{sC_{1}R_{x1} }{sC_{1}R_{x1}+1}\bigg{)}V_{in} \tag{12}\]  
\[\text{LPF}\qquad\quad I_{o4}=-I_{o3}=\frac{1}{R_{in}}\bigg{(}\frac{1}{sC_{1}R_ {x1}+1}\bigg{)}V_{in} \tag{13}\]  
\[\text{APF}+\qquad\quad\quad I_{AP+}=\frac{1}{R_{in}}\big{(}I_{o4}+I_{o1}\big{)} =\frac{1}{R_{in}}\Big{(}\frac{1-sC_{1}R_{x1}}{sC_{1}R_{x1}}\Big{)}V_{in} \tag{14}\]  
\[\text{APF}-\qquad\quad I_{AP-}=\frac{1}{R_{in}}(I_{o2}+I_{o3})=\frac{1}{R_{in }}\bigg{(}\frac{sC_{1}R_{x1}-1}{sC_{1}R_{x1}+1}\bigg{)}V_{in} \tag{15}\]  
Figure 4c shows the proposed VM filter. In this case, the resistor \(R_{in}\) is used to convert the voltage input \(V_{in}\) to the current input \(I_{in}\) and the resistors \(R_{out}\) are used to convert the current outputs \(I_{o1}\), \(I_{o2}\), \(I_{o3}\), and \(I_{o4}\) to the voltage outputs \(V_{o1}\), \(V_{o2}\), \(V_{o3}\), \(V_{o4}\), \(V_{AP+}\), and \(V_{AP-}\). Using the transfer functions (3)-(6) of the current-mode filter, the output voltages of the VM filter can be expressed by  
\[\text{HPF}\qquad\quad V_{o2}=-V_{o1}=\frac{R_{out}}{R_{in}}\bigg{(}\frac{sC_{1 }R_{x1}}{sC_{1}R_{x1}+1}\bigg{)}V_{in} \tag{16}\]  
\[\text{LPF}\qquad\quad V_{o4}=-V_{o3}=\frac{R_{out}  
Considering the nonidealities of the EX-CCCII, the output currents \(I_{01}\), \(I_{02}\), \(I_{03}\), and \(I_{04}\) can be rewritten as  
\[I_{01}=-I_{02}=\left(\frac{sC_{1}R_{x1}\beta_{2-}+(1-\beta_{1-})\beta_{2-}}{sC_{ 1}R_{x1}+(1-\beta_{1-})+\beta_{1+}\beta_{2-}}\right)I_{in} \tag{21}\]  
\[I_{03}=-I_{04}=\left(\frac{\beta_{1+}\beta_{2-}}{sC_{1}R_{x1}+(1-\beta_{1-})+ \beta_{1+}\beta_{2-}}\right)I_{in} \tag{22}\]  
From (21) and (22), the current tracking error (i.e., \(\beta_{1-}\)) will cause a slight deviation from ideal characteristics in the HPF and LPF transfer functions. Letting \(\beta_{1-}=1\), (21) and (22) can be rewritten as  
\[I_{01}=-I_{02}=\left(\frac{sC_{1}R_{x1}\beta_{2-}}{sC_{1}R_{x1}+\beta_{1+} \beta_{2-}}\right)I_{in} \tag{23}\]  
\[\text{LPF}\qquad\quad V_{03}=-V_{04}=R_{out}\bigg{(}\frac{1}{sC_{1}R_{x1}+1} \bigg{)}\,I_{in} \tag{9}\]0  
The output currents of the APF become  
\[\text{LPF}\qquad\quad V_{03}=-V_{04}=R_{out}\bigg{(}\frac{1}{sC_{1}R_{x1}+1} \bigg{)}\,I_{in} \tag{9}\]1  
\[\text{LPF}\qquad\quad V_{03}=-V_{04}=R_{out}\bigg{(}\frac{1}{sC_{1}R_{x1}+1} \bigg{)}\,I_{in} \tag{9}\]2  
The pole frequency of all filters can be rewritten as  
\[\text{LPF}\qquad\quad V_{03}=-V_{04}=R_{out}\bigg{(}\frac{1}{sC_{1}R_{x1}+1} \bigg{)}\,I_{in} \tag{9}\]3  
Using the results for the EX-CCCII that include parasitic impedances in [50], the capacitance \(C_{1}\) can be selected as \(C_{1}>(C_{z1-}+C_{z2-})\) to avoid parasitic capacitance effects to the pole frequency, where \(C_{z1-}\) and \(C_{z2-}\) are the parasitic capacitances at the z\({}_{1-}\)- and z\({}_{2-}\)-terminals, respectively.