## 6 Parasitic Element Analysis
### Parasitic Effects on CM Inverse Filter Realization  
The non-ideal transfer functions of the proposed CM first-order inverse filter in Figure 3a are derived by accounting for the parasitic nature of the VCII-:  
\[1\text{st}-\text{order ILP}:\ \ G_{i1}(s)\ =\ \frac{G_{0}(sR_{1}C_{1}+1)}{(sR_{1}C_{x1 }+1)\big{[}s(R_{x1}+R_{y2})C_{A}\big{]}}, \tag{17}\]  
\[1\text{st}-\text{order IHP}:\ \ G_{i1}(s)\ =\ \frac{G_{0}(sR_{1}C_{1}+1)}{(sR_{1}C_{1})( sR_{1}C_{x1}+1)}. \tag{18}\]If \(f_{i1}\ =\ \frac{1}{2\pi R_{1}c_{x1}}\) and \(f_{i2}\ =\ \frac{1}{2\pi\left(R_{z1}+R_{yz}\right)C_{A}}\), then the frequency limitations of the proposed first-order CM ILP and IHP filters should lie in the following ranges:  
\[1\text{st-order ILP:}\ \ \ \ \ \ \ \ \ \ \ \ f\ <<\min\left[f_{i1},f_{i2}\right], \tag{19}\]  
\[1\text{st-order IHP:}\ \ \ \ \ \ \ \ \ f\ <<\min\left[f_{i1}\right]. \tag{20}\]  
Furthermore, we have also considered the parasitic effect on the proposed CM second-order inverse filter structure in Figure 3b. The various non-ideal expressions of ILP, IHP, and IBP filter responses are as follows:  
\[2\text{nd-order ILP:}\] \[G_{i2}(s)\ =\ \frac{G_{0}N(s)}{(sR_{1}C_{x1}+1)(sR_{2}C_{x2}+1) \left[s(R_{x1}+R_{yz2})C_{A}+1\right]\left[s\left(R_{z2}+R_{yz3}  
Figure 5: CMOS implementation of VCII-. Figure 6: Time and frequency responses of the first-order VM inverse filter in Figure 2a. (**a**) ILP responses; (**b**) IHP responses. Figure 7: Time and frequency responses of the second-order VM inverse filter in Figure 2b. **(a)** ILP responses; **(b)** IHP responses; **(c)** IBP responses. The simulated cut-off frequency \(f_{0}\) for the first-order VM ILP and IHP responses was measured at 1.56 MHz and 1.54 MHz, respectively, and the corresponding \(f_{0}\) values for the second-order ILP, IHP, and IBP responses were 1.57 MHz, 1.52 MHz, and 1.55 MHz. For the CM inverse filters, the simulated \(f_{0}\) values of the first-order ILP and IHP responses were 1.57 MHz and 1.50 MHz, while the second-order ILP, IHP, and IBP filters provided the simulated \(f_{0}\) at 1.57 MHz, 1.52 MHz, and 1.56 MHz, respectively. The \(f_{0}\)-tuning for the IBP filter in Figure 2b was simulated with a constant value of \(Q\). The capacitor and resistor values are \(C_{1}=C_{2}=C_{A}=100\) pF, \(R_{B}=1\) k\(\Omega\), and \(R_{1}=R_{2}=0.5\) k\(\Omega\), 2 k\(\Omega\), and 5 k\(\Omega\). With a fixed \(Q\)-value of 0.5, the corresponding values for \(f_{0}\) are 3.18 MHz, 795.77 MHz, and 318.31 MHz, respectively. Figure 10 illustrates the ideal and simulated gain-frequency responses, demonstrating the ability to independently adjust the \(f_{0}\)-value of the proposed filter. The proposed VM IBP filter in Figure 2b was also simulated at different temperatures, including 0 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, 75 \({}^{\circ}\)C, and 100 \({}^{\circ}\)C. The simulations were conducted with \(C_{1}=C_{2}=C_{A}=100\) pF and \(R_{1}=R_{2}=R_{B}=1\) k\(\Omega\), and the obtained results are shown in Figure 11. It is observed that the gain-frequency response of the suggested circuit is found to increase with temperature. The working transistor current reduces with increasing temperature, whilst the values of resistors and capacitors increase, leading to an increase in gain. In this instance, temperature variations influence the gain response of the proposed inverse filter, as it is dependent upon the currents in the transistor  
Figure 8: Time and frequency responses of the first-order CM inverse filter in Figure 3a. (**a**) ILP responses; (**b**) IHP responses. devices and the values of the resistors and capacitors. At \(f_{0}\) = 1.59 MHz, the deviations from its nominal value in voltage gain due to temperature variations are approximately \(-0.88\) dBV at 0 \({}^{\circ}\)C and +7.10 dBV at 100 \({}^{\circ}\)C, which can be considered insignificant.