## 4 Non-Ideal Analysis  
The mismatch between current mirrors, process variation and the device mismatch between the MOS transistors results in variation in the frequency dependent voltage and current transfer gains of the VDBA. This change in the gain from the ideal value of unity results in the deviation in the center frequency and quality factor of the filter. The effect of frequency dependent non-deal voltage and current transfer gains is analyzed in this section. The (\(\beta\)) denotes the frequency dependent non-ideal voltage transfer gain and (\(\gamma\)) represents the transconductance transfer gain of the OTA. Considering the effect of the non-ideal gains the V-I characteristics of the VDBA will be modified as given in Equations (16)-(18), where \(\beta_{m}=1-\epsilon_{vm}\) and \(\gamma_{m}=1-\epsilon_{\text{gm}}\), for \(m=1,2\), which refers to the number of VDBAs. Here, \(\epsilon_{vm}(|\epsilon_{vm}|~{}\ll~{}1)\) denote voltage tracking error, and \(\epsilon_{\text{gm}}(|\epsilon_{\text{gm}}|~{}\ll~{}1)\) denote transconductance errors of the VDBA. The modified transfer function and expressions for center frequency and quality factor including the non-ideal effect is given in Equations (19)-(24) for the MISO filter. The non-idealities result in deviations from the expected value. \[I_{\text{Z}}=I_{\text{ZC+}}=\gamma g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{16}\]  
\[I_{\text{ZC-}}=\gamma^{\prime}g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{17}\]  
\[V_{\text{Z}}=\beta_{m}V_{\text{W}}. \tag{18}\]  
\[V_{\text{OUT(VM)}}=\beta_{1}\Bigg{[}\frac{s^{2}C_{1}C_{2}V_{2}-sC_{1}g_{\text{ m1}}\gamma_{1}V_{3}+g_{\text{m1}}g_{\text{m3}}\gamma_{1}\gamma_{3}V_{1}}{s^{2}C_{1}C_{2}+sC _{1}g_{\text{m2}}+\gamma_{1}\gamma_{3}g_{\text{m1}}g_{\text{m3}}}\Bigg{]}, \tag{19}\]  
\[I_{\text{OUT(CM)}}=\Bigg{[}\frac{(s^{2}C_{1}C_{2}+sC_{1}g_{\text{m2}}+\gamma_{ 1}\gamma_{3}g_{\text{m1}}g_{\text{m3}})I_{3}-sC_{1}g_{\text{m2}}\gamma_{2}g_{1 }I_{2}-\gamma_{1}\gamma_{2}g_{1}g_{\text{m1}}g_{\text{m2}}I_{1}}{s^{2}C_{1}C_{ 2}+sC_{1}g_{\text{m2}}+\gamma_{1}\gamma_{3}g_{\text{m1}}g_{\text{m3}}}\Bigg{]}, \tag{20}\]  
\[I_{\text{OUT(TAM)}}=-\gamma_{2}\beta_{1}g_{\text{m2}}\Bigg{[}\frac{s^{2}C_{1} C_{2}V_{2}-sC_{1}g_{\text{m1}}\gamma_{1}V_{3}+g_{\text{m1}}g_{\text{m3}} \gamma_{1}\gamma_{3}V_{1}}{s^{2}C_{1}C_{2}+sC_{1}g_{\text{m2}}+\gamma_{1}\gamma _{3}g_{1}g_{\text{m3}}}\Bigg{]}, \tag{21}\]  
\[V_{\text{OUT(TIM)}}=R_{1}\Bigg{[}\frac{(s^{2}C_{1}C_{2}+sC_{1}g_{\text{m2}}+ \gamma_{1}\gamma_{3}g_{\text{m1}}g_{\text{m3}})I_{4}-sC_{1}g_{\text{m2}}\gamma _{2}g_{1}I_{2}-\gamma_{1}\gamma_{2}g_{1}g_{\text{m1}}g_{\text{m2}}I_{1}}{s^{2} C_{1}C_{2}+sC_{1}g_{\text{m2}}+\gamma_{1}\gamma_{3}g_{\text{m1}}g_{\text{m3}}} \Bigg{]}, \tag{22}\]  
\[\omega_{0}=\sqrt{\frac{\gamma_{1}\gamma_{3}g_{\text{m1}}g_{\text{m3}}}{C_{1}C_{ 2}}}, \tag{23}\]  
\[Q=\frac{1}{g_{\text{m2}}}\sqrt{\frac{\gamma_{1}\gamma_{3}g_{\text{m1}}g_{\text{ m3}}C_{2}}{C_{1}}}. \tag{24}\]  
The non-ideal expressions of the filter transfer function for the CM SIMO filter are given as follows:  
\[\frac{I_{\text{LP}}}{I_{\text{IN}}}=\frac{\gamma_{1}\gamma_{3}g_{\text{m1}}g_{ \text{m3}}}{s^{2}C_{1}C_{2}+sC_{1}g_{\text{m2}}+\gamma_{1}\gamma_{3}g_{\text{m1 }}g_{\text{m3}}}, \tag{25}\]  
\[I_{\text{ZC-}}=\gamma^{\prime}g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{17}\]0  
\[I_{\text{ZC-}}=\gamma^{\prime}g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{17}\]1The sensitivities of \(\omega_{0}\) and \(Q\) with respect to the non-ideal gain and passive elements are calculated. Equations (28)-(30) indicate identical sensitivities for the MISO mixed-mode universal filter and CM SIMO filter. \[I_{\text{ZC-}}=\gamma^{\prime}g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{17}\]2  
\[I_{\text{ZC-}}=\gamma^{\prime}g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{17}\]3  
\[I_{\text{ZC-}}=\gamma^{\prime}g_{\text{m}}(V_{\text{P}}-V_{\text{N}}), \tag{17}\]4  
The sensitivities are not greater than one, which is desired.