## 6 Parasitic Element Analysis
### Parasitic Effects on VM Inverse Filter Realization  
A straightforward analysis of the proposed first-order VM inverse filter in Figure 2a using the VCII's non-ideal model in Figure 4 yields the non-ideal transfer functions of the ILP and IHP filters given by the following expressions:  
\[1\text{st}-\text{order ILP}:\qquad\ G_{v1}(s)\ =\ \frac{G_{0}(sR_{1}C_{1}+1)}{(sR_{1}C_{x} +1)\big{(}sR_{y}C_{A}+1\big{)}}, \tag{7}\]  
\[1\text{st}-\text{order IHP}:\qquad\ G_{v1}(s)\ =\ \frac{G_{0}(sR_{1}C_{1}+1)}{(sR _{1}C_{1})(sR_{1}C_{x}+1)}. \tag{8}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter Type** & \begin{tabular}{c} **Transfer Function,** \\ \(\mathbf{G_{2}}(s)\)**=\(\frac{Lm(s)}{Lm(s)}\) \\ \end{tabular} & \begin{tabular}{c} **Minimum Gain** \\ **(\(\mathbf{G_{0}}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Cut-Off** \\ **Frequency** \\ **(\(\mathbf{\omega_{0}}\))** \\ \end{tabular} &
\begin{tabular}{c} **Quality** \\ **Factor** \\ **(Q)** \\ \end{tabular} \\ \hline ILP & \(G_{0}N(s)\) & \(\frac{\alpha_{142}\beta_{1}\beta_{2}\beta_{3}C_{A}C_{B}}{C_{1}C_{2}}\) & & \\ IHP & \(G_{0}\Big{[}\frac{N(s)}{s(R_{1}C_{1}+R_{2}C_{2})}\Big{]}\) & \(\frac{\alpha_{142}\beta_{1}\beta_{2}\beta_{3}C_{A}}{K_{A}R_{B}}\) & \(\frac{1}{\sqrt{R_{1}R_{2}C_{1}C_{2}}}\) & \(\frac{\sqrt{R_{1}R_{2}C_{1}C_{2}}}{R_{1}C_{1}+R_{2}C_{2}}\) \\ IBP & \(G_{0}\Big{[}\frac{N(s)}{s(R_{1}C_{1}+R_{2}C_{2})}\Big{]}\) & \((\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}\beta_{3})\Big{(}\frac{R_{1}}{C_{2}}+ \frac{R_{1}}{C_{1}}\Big{)}\Big{(}\frac{C_{A}}{R_{B}}\Big{)}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Non-ideal parameters of the second-order CM inverse filter in Figure 3b. Figure 4: Non-ideal equivalent circuit for the VCII with its parasitis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Filter Type** & \begin{tabular}{c} **Transfer Function,** \\ \(\mathbf{G_{\Pi}}(s)\)**=\(\frac{Lm(s)}{Lm(s)}\) \\ \end{tabular} & \
\begin{tabular}{c} **Minimum Gain** \\ **(\(\mathbf{G_{0}}\))** \\ \end{tabular} &
\begin{tabular}{c} **Cut-Off Frequency** \\ **(\(\mathbf{\omega_{0}}\))** \\ \end{tabular} \\ \hline ILP & \(G_{0}(sR_{1}C_{1}+1)\) & \(\frac{\alpha_{1}\beta_{2}\beta_{3}C_{A}}{C_{1}C_{2}}\) & \(\frac{1}{K_{1}C_{1}}\) \\ IHP & \(G_{0}\Big{(}\frac{sR_{1}C_{1}+1}{sR_{1}C_{1}}\Big{)}\) & \(\frac{\alpha_{1}\beta_{2}\beta_{3}R_{1}}{K_{A}R_{B}}\) & \(\frac{1}{K_{1}C_{1}}\) \\ IBP & \(G_{0}\Big{[}\frac{N(s)}{s(R_{1}C_{1}+R_{2}C_{2})}\Big{]}\) & \((\alpha_{1}\alpha_{2}\beta_{1}\beta_{2}\beta_{3})\Big{(}\frac{R_{1}}{C_{2}}+ \frac{R_{1}}{C_{1}}\Big{)}\Big{(}\frac{C_{A}}{R_{B}}\Big{)}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Non-ideal parameters of the first-order CM inverse filter in Figure 3a. Typically, the value of \(R_{x}\) is extremely high, while the values of \(R_{y}\) and \(C_{x}\) are extremely small. As a consequence, \(R_{x}>>R_{1}\), \(R_{A}>>R_{y}\), and \(C_{x}<<C_{1}\). Equations (7) and (8) demonstrate that the VCII parasitics introduce extra poles at frequencies \(f_{v1}=(1/2\pi R_{1}C_{x})\) and \(f_{v2}=(1/2\pi R_{y}C_{A})\) for the ILP filter response, and at frequency \(f_{v1}\) in IHP. Therefore, the useful frequencies of the first-order VM ILP and IHP filters are approximated as follows:  
\[1\text{st-order ILP:}\qquad\qquad f<<\min\big{[}f_{v1},f_{v2}\big{]}, \tag{9}\]  
\[1\text{st-order IHP:}\qquad\qquad f<<\min\big{[}f_{v1}\big{]}. \tag{10}\]  
For instance, if the parasitic elements of the VCII- are \(R_{y}=38.88\)\(\Omega\), \(R_{x}=109.24\) k\(\Omega\), \(R_{z}=38.23\)\(\Omega\), and \(C_{x}=3\) pF [32], and the filter components are chosen as \(R_{1}=1\) k\(\Omega\) and \(C_{A}=100\) pF, then the pole frequencies will be located at frequencies \(f_{v1}=53.08\) MHz and \(f_{v2}=40.93\) MHz. This implies that the first-order VM ILP and IHP filters proposed here have high frequency limitations of approximately 40.93 MHz and 53.05 MHz, respectively.