## 2 Circuit Descriptions
### Effect of the CCCCTA Parasitic Impedances and Design Considerations  
Next, various parasitic impedances of the CCCCTA in the proposed circuit were studied. Figure 3 represents the non-ideal CCCCTA model including its parasitic elements. A port Y parasitic is in the form \(\mathrm{R_{Yi}}//\mathrm{C_{Yi}}\), a port Z+ parasitic is in the form \(\mathrm{R_{Zi+}}//\mathrm{C_{Zi+}}\), a port Z- parasitic is in the form \(\mathrm{R_{Zi-}}//\mathrm{C_{Zi-}}\), a port O1 parasitic is in the form \(\mathrm{R_{Oii}}//\mathrm{C_{Oii}}\), a port O2 parasitic is in the form \(\mathrm{R_{O2i}}//\mathrm{C_{O2i}}\), a port -O1 parasitic is in the form \(\mathrm{R_{Oii}}//\mathrm{C_{Oii}}\), a port -O2 parasitic is in the form \(\mathrm{R_{O2i}}//\mathrm{C_{O2i}}\), and a port X parasitic is in the form \(\mathrm{R_{Xi}}\) where i = 1, 2, 3 and indicates the ith CCCCTA. After applying the non-ideal equivalent circuit mode of the CCCCTA in the proposed circuit, the denominator of the transfer functions becomes:  
\[\mathrm{D(s)}=\frac{\mathrm{m}}{\mathrm{R_{3p}}}\mathrm{s}^{2}+\mathrm{n_{2}} \mathrm{g_{m1}}\mathrm{s}+\mathrm{n_{1}}\mathrm{n_{2}}\mathrm{g_{m2}} \tag{40}\]  
where,  
\[\mathrm{n_{1}}=(\frac{1}{\mathrm{R_{X2}}})\frac{\mathrm{sR_{1p}}}{1+\mathrm{ sR_{1p}C_{1}^{\prime}}}=(\frac{1}{\mathrm{R_{X2}C_{1}^{\prime}}})(\frac{ \mathrm{s}}{\mathrm{s}+\omega_{1}}),\ \omega_{1}=\frac{1}{\mathrm{R_{1p}C_{1}^{ \prime}}} \tag{41}\]  
\[\mathrm{n_{2}}=(\frac{1}{\mathrm{R_{X1}}})\frac{\mathrm{sR_{2p}}}{1+\mathrm{ sR_{2p}C_{2}^{\prime}}}=(\frac{1}{\mathrm{R_{X1}C_{2}^{\prime}}})(\frac{ \mathrm{s}}{\mathrm{s}+\omega_{2}}),\ \omega_{2}=\frac{1}{\mathrm{R_{2p}C_{2}^{ \prime}}} \tag{42}\]  
\[\mathrm{m}=1+\mathrm{sR_{3p}C_{3p}}=\frac{1}{\omega_{3}}(\mathrm{s}+\omega_{3} ),\ \omega_{3}=\frac{1}{\mathrm{R_{3p}C_{3p}}} \tag{43}\]  
\(\mathrm{C_{1}^{\prime}}=\mathrm{C_{1}}//\mathrm{C_{Z2+}}\), \(\mathrm{C_{2}^{\prime}}=\mathrm{C_{2}}//\mathrm{C_{Z1+}}//\mathrm{C_{Y2}}\), \(\mathrm{C_{3p}}=\mathrm{C_{O11}}//\mathrm{C_{O12}}//\mathrm{C_{Z3-}}\), \(\mathrm{R_{1p}}=\mathrm{R_{Z2+}}\), \(\mathrm{R_{2p}}=\mathrm{R_{Z1+}}//\mathrm{R_{Y2}}\), and \(\mathrm{R_{3p}}=\mathrm{R_{X1}}//\mathrm{R_{O11}}//\mathrm{R_{O12}}//\mathrm{ R_{Z3-}}\). Equations (41)-(43) illustrate that the effects of parasitic elements depend on three parasitic poles yielded by the non-idealities of CCCCFAs. For near-ideal frequency operation, the operating frequency must be higher than \(\omega_{1}\) and \(\omega_{2}\) and lower than \(\omega_{3}\). Therefore, the useful frequency range of the proposed filter is limited by the following conditions:  
\[10\times\mathrm{max}\{\omega_{1},\omega_{2}\}<<\omega<<0.1\omega_{3} \tag{44}\]  
This condition is easily satisfied since the external capacitance can be set much higher than the parasitic capacitance. In Figure 2, the effects of CCCCTA parasitic elements on the proposed filter can be ignored under the following conditions: \(\mathrm{min}\) (\(\mathrm{C_{1}}\), \(\mathrm{C_{2}}\)) >> parasitic capacitances (\(\mathrm{C_{Z1+}}\), \(\mathrm{C_{22+}}\), \(\mathrm{C_{Y2}}\)), parasitic resistances (\(\mathrm{R_{O11}}\), \(\mathrm{R_{O12}}\), \(\mathrm{R_{Z3-}}\)) >> \(\mathrm{R_{X1}}\), \(1/\mathrm{sC_{1}}<<\mathrm{R_{Z2+}}\) and \(1/\mathrm{sC_{2}}<<\mathrm{R_{Z1+}}//\mathrm{R_{Y2}}\). Figure 3: Non-ideal equivalent circuit model of the CCCCTA. According to (44), the effects of parasitic elements on coefficients n\({}_{1}\), n\({}_{2}\) and m diminish under the conditions \(|\)s\(|\)\(>>\)\(\omega_{1}\), \(|\)s\(|\)\(>>\)\(\omega_{2}\) and \(|\)s\(|\)\(<<\)\(\omega_{3}\).