## 2 Circuit Descriptions
### Proposed Electronically-Tunable Mixed-Mode Biquadratic Filter
#### 2.2.1 CM and TIM  
According to Equations (2)-(10), CM and TIM can be obtained by setting the input voltage \(V_{in}=0\) (grounded) and getting \(I_{in}\) as input signal. The five current transfer functions obtained are:  
\[\frac{I_{o1}}{I_{in}}=\frac{\mathrm{s^{2}C_{1}C_{2}G_{X1}}}{\mathrm{D(s)}} \tag{12}\]  
Figure 2: Proposed electronically tunable mixed-mode biquadratic filter. \[\frac{\mathrm{I_{o2}}}{\mathrm{I_{in}}}=\frac{-\mathrm{sC_{1}g_{m1}G_{X1}}}{ \mathrm{D(s)}} \tag{13}\]  
\[\frac{\mathrm{I_{o3}}}{\mathrm{I_{in}}}=\frac{\mathrm{g_{m2}G_{X1}G_{X2}}}{ \mathrm{D(s)}} \tag{14}\]  
\[\frac{\mathrm{I_{od}}}{\mathrm{I_{in}}}=\frac{\mathrm{I_{o5}}}{\mathrm{I_{in}}}= \frac{\mathrm{sC_{1}g_{m1}G_{X1}}}{\mathrm{D(s)}} \tag{15}\]  
Equations (12)-(15) indicate that a non-inverting HP filtering response is obtained from \(\mathrm{I_{o1}}\), an inverting BP filtering response is obtained from \(\mathrm{I_{o2}}\), a non-inverting LP filtering response is obtained from \(\mathrm{I_{o3}}\), and two non-inverting BP filtering responses are obtained from \(\mathrm{I_{od}}\) and \(\mathrm{I_{o5}}\). The BS filtering response is easily obtained by adding the two currents \(\mathrm{I_{o1}}\) and \(\mathrm{I_{o3}}\) to obtain the following transfer function:  
\[\frac{\mathrm{I_{BS}}}{\mathrm{I_{in}}}=\frac{\mathrm{I_{o1}}+\mathrm{I_{o3}}} {\mathrm{I_{in}}}=\frac{\mathrm{s}^{2}\mathrm{C_{1}C_{2}G_{X1}}+\mathrm{g_{m2 }G_{X1}G_{X2}}}{\mathrm{D(s)}} \tag{16}\]  
Similarly, the AP transfer function is easily obtained by adding the three currents \(\mathrm{I_{o1}}\), \(\mathrm{I_{o2}}\) and \(\mathrm{I_{o3}}\) to obtain the following transfer function:  
\[\frac{\mathrm{I_{AP}}}{\mathrm{I_{in}}}=\frac{\mathrm{I_{o1}}+\mathrm{I_{o2}} +\mathrm{I_{o3}}}{\mathrm{I_{in}}}=\frac{\mathrm{s}^{2}\mathrm{C_{1}C_{2}G_{X1 }}-\mathrm{sC_{1}g_{m1}G_{X1}}+\mathrm{g_{m2}G_{X1}G_{X2}}}{\mathrm{D(s)}} \tag{17}\]  
Thus, all five standard filtering responses are provided by the same CM biquadratic filter structure, and no other component-matching conditions are needed. Notably, all current outputs are available from high-output impedance terminals. High-output impedance terminals of the configuration enable the circuit to be cascaded without additional current buffers. Because the gain of the current-mode HP, LP and BP filters is unity, the additional current amplifiers are needed if a variable gain of current-mode filter is necessary. In addition, the BS and AP filters cannot be realised simultaneously with the HP, LP and BP filters. This problem can be solved by adding the multiple current outputs that can be easily implemented by simply adding output branches. According to Equation (11), filter parameters \(\omega_{\mathrm{o}}\) and \(\mathrm{Q}\) are:  
\[\omega_{\mathrm{o}}=\sqrt{\frac{\mathrm{g_{m2}G_{X2}}}{\mathrm{C_{1}C_{2}}}}, \ \mathrm{Q}=\frac{1}{\mathrm{g_{m1}}}\sqrt{\frac{\mathrm{g_{m2}G_{X2}C_{2}}}{ \mathrm{C_{1}}}} \tag{18}\]  
Based on Equation (18), parameter \(\mathrm{Q}\) can be independently tuned by using \(\mathrm{g_{m1}}\) without disturbing \(\omega_{\mathrm{o}}\). Restated, parameters \(\omega_{\mathrm{o}}\) and \(\mathrm{Q}\) are orthogonally adjustable through the finite input conductance \(\mathrm{G_{X2}}\) and then the \(\mathrm{g_{m1}}\) in that order. This property is desirable in biquadratic filters because it increases design and tuning flexibility. Accordingly, the four TIM transfer functions in this case can be obtained as follows:  
\[\frac{\mathrm{V_{o1}}}{\mathrm{I_{in}}}=\frac{-\mathrm{G_{X1}G_{X2}}}{ \mathrm{D(s)}} \tag{19}\]  
\[\frac{\mathrm{V_{o2}}}{\mathrm{I_{in}}}=\frac{-\mathrm{sC_{1}G_{X1}}}{ \mathrm{D(s)}} \tag{20}\]  
Figure 2:0  
Figure 2:1  
As indicated by Equations (19)-(22), an inverting LP filtering response is obtainedfrom V\({}_{03}\), and a non-inverting BP filtering response is obtained from V\({}_{04}\). That is, the circuit provides TIM HP, LP and two BP responses simultaneously without disturbing circuit topology.