# Universal Voltage Conveyor and its Novel Dual-Output Fully-Cascadable VM APF Application
## 3 Non-Ideal Analysis  
For a complete analysis of the circuit, it is also important consider the main non-idealities of the UVC in Equation (1) as also shown in Figure 2, where:  
* the parasitic resistance \(R_{\rm X}\) and parasitic capacitance \(C_{\rm X}\) appear between the high-impedance terminal X of the UVC and ground and their values computed in SPICE software are \(R_{\rm X}=378.73\) k\(\Omega\parallel C_{\rm X}=17.41\) pF, respectively,
* the non-zero parasitic resistance \(R_{\rm YP}\) and \(R_{\rm YN}\) at two difference current inputs YP and YN have values \(R_{\rm YP}=1.27\)\(\Omega\) and \(R_{\rm YN}=0.51\)\(\Omega\), respectively,
* the parasitic resistance \(R_{\rm W}\) and parasitic capacitance \(C_{\rm W}\) appear between the auxiliary terminal W of the UVC and ground and their values are \(R_{\rm W}=88.19\) M\(\Omega\parallel C_{\rm W}=4.19\) pF, respectively,
* the non-zero parasitic resistance \(R_{\rm ZP}\) and \(R_{\rm ZN}\) at mutually inverse voltage outputs ZP and ZN have values \(R_{\rm ZP}=1.01\)\(\Omega\) and \(R_{\rm ZN}=0.71\)\(\Omega\), respectively.  
These parameters can be found in greater detail in [22; 23; 24]. Now, considering the non-idealities listed above, the proposed VM APF suffers from the following non-idealities:  
* at the node 1, a parasitic impedance \(Z_{\rm X}=1/Y_{\rm X}=R_{\rm X}\parallel(1/sC_{\rm X})\) can be seen. Note that the capacitance \(C_{\rm X}\) and resistance \(R_{\rm X}\) can be absorbed into external capacitor \(C_{1}\) and resistor \(R_{1}\), respectively, as they appear in parallel. Hence, in further analysis the total capacitance and resistance at this node will be considered \(C_{1}^{\prime}\) and \(R_{1}^{\prime}=Z_{1}^{\prime}\),
* the node 2 at low-impedance terminal YP is characterized by a parasitic impedance \(Z_{\rm YP}=R_{\rm YP}\), which is in series with external capacitor \(C_{2}\), and by assuming it as an impedance \(Z_{C_{2}}=1/Y_{C_{2}}=1/sC_{2}\), the total impedance at this node can be described as \(Z_{C_{2}}^{\prime}\),
* finally, the node 3 can be characterized by a parasitic impedance \(Z_{\rm YN}=R_{\rm YN}\), which appears in series with external resistor \(R_{2}=Z_{2}\). Hence, the total impedance at this node can be labeled as \(Z_{2}^{\prime}\)  
Figure 2: Main parasitic resistance and capacitance of the UVC.  
Considering the non-ideal current and voltage gains of the UVC and simultaneously the effect of the aforementioned non-idealities and re-analyzing the proposed VM APF, the ideal TFs (4) convert to:  
\[T_{1}^{\prime}\left(s\right)=\frac{\gamma_{\mathrm{01}}Z_{1}^{\prime}\left( \alpha_{\mathrm{01}}\delta_{\mathrm{01}}Z_{2}^{\prime}-\alpha_{\mathrm{02}} \delta_{\mathrm{02}}Z_{C_{2}}^{\prime}\right)}{Z_{2}^{\prime}Z_{C_{2}}^{\prime }\left(sC_{1}^{\prime}Z_{1}^{\prime}+1\right)}\,,\ \ \ \ \ \ T_{2}^{\prime}\left(s\right)=-\frac{\gamma_{\mathrm{02}}}{ \gamma_{\mathrm{01}}}.T_{1}^{\prime}\left(s\right). \tag{6}\]  
Subsequently, the non-ideal zero and pole frequencies differ and can be found as:  
\[\omega_{\mathrm{z}}=\frac{\alpha_{\mathrm{02}}\delta_{\mathrm{02}}}{\alpha_{ \mathrm{01}}\delta_{\mathrm{01}}}.\frac{Z_{C_{2}}^{\prime}}{Z_{2}^{\prime}}, \ \ \ \ \ \omega_{\mathrm{p}}=\frac{1}{C_{1}^{\prime}Z_{1}^{\prime}}. \tag{7}\]  
The effect of non-idealities on the proposed VM APF can be significantly minimized by the proper selection of external passive components and/or by precise design of the UVC.