## 2 The Proposed Isomorphic Circuits
Routine analysis of the VM NBPF circuit (Figure 2a) yielded the following three NBPF responses:  
\[\frac{\mathrm{V}_{\mathrm{BPF1}}}{\mathrm{V}_{\mathrm{in}}}=\frac{\mathrm{sC}_{ \mathrm{1}}\mathrm{g}_{\mathrm{m2}}\mathrm{R}_{\mathrm{1}}}{\mathrm{s}^{2} \mathrm{C}_{\mathrm{1}}\mathrm{C}_{\mathrm{2}}\mathrm{R}_{\mathrm{1}}+\mathrm{ sC}_{\mathrm{1}}+\mathrm{g}_{\mathrm{m1}}\mathrm{g}_{\mathrm{m3}}\mathrm{R}_{ \mathrm{1}}} \tag{10}\]  
\[\frac{\mathrm{V}_{\mathrm{BP2}}}{\mathrm{V}_{\mathrm{in}}}=\frac{\mathrm{sC}_{ \mathrm{1}}\mathrm{g}_{\mathrm{m2}}\mathrm{R}_{\mathrm{1}}}{\mathrm{s}^{2} \mathrm{C}_{\mathrm{1}}\mathrm{C}_{\mathrm{2}}\mathrm{R}_{\mathrm{1}}+\mathrm{ sC}_{\mathrm{1}}+\mathrm{g}_{\mathrm{m1}}\mathrm{g}_{\mathrm{m3}}\mathrm{R}_{ \mathrm{1}}} \tag{11}\]  
\[\frac{\mathrm{V}_{\mathrm{BP3}}}{\mathrm{V}_{\mathrm{in}}}=(\frac{\mathrm{sC}_{ \mathrm{1}}\mathrm{g}_{\mathrm{m2}}\mathrm{R}_{\mathrm{1}}}{\mathrm{s}^{2} \mathrm{C}_{\mathrm{1}}\mathrm{C}_{\mathrm{2}}\mathrm{R}_{\mathrm{1}}+\mathrm{ sC}_{\mathrm{1}}+\mathrm{g}_{\mathrm{m1}}\mathrm{g}_{\mathrm{m3}}\mathrm{R}_{ \mathrm{1}}})(1+\frac{\mathrm{R}_{\mathrm{2}}}{\mathrm{R}_{\mathrm{3}}}) \tag{12}\]  
Based on the three NBPF responses from Equations (10)-(12), the NBPF parameters of \(\mathrm{f}_{\mathrm{o}}\) and \(\mathrm{Q}\) are given by  
\[\mathrm{f}_{\mathrm{o}}=\frac{1}{2\pi}\sqrt{\frac{\mathrm{g}_{\mathrm{m1}} \mathrm{g}_{\mathrm{m3}}}{\mathrm{C}_{\mathrm{1}}\mathrm{C}_{\mathrm{2}}}} \tag{13}\]  
\[\mathrm{Q}=\mathrm{R}_{\mathrm{1}}\sqrt{\frac{\mathrm{C}_{\mathrm{2}}\mathrm{ g}_{\mathrm{m1}}\mathrm{g}_{\mathrm{m3}}}{\mathrm{C}_{\mathrm{1}}}} \tag{14}\]  
From Equations (13) and (14), the NBPF parameters of \(\mathrm{f}_{\mathrm{o}}\) and \(\mathrm{Q}\) have orthogonal tunability by tuning the \(\mathrm{g}_{\mathrm{m1}}\) for \(\mathrm{f}_{\mathrm{o}}\) first and then the grounded resistor \(\mathrm{R}_{\mathrm{1}}\) for \(\mathrm{Q}\) without affecting parameter \(\mathrm{f}_{\mathrm{o}}\). Figure 3: The second proposed isomorphic circuit of the VM NBPF and VM QSO. (**a**) VM NBPF circuit. (**b**) VM QSO circuit. Figure 2: The first proposed isomorphic circuit of the VM NBPF and VM QSO. (**a**) VM NBPF circuit. (**b**) VM QSO circuit. The passband gains of the three NBPF responses are given by  
\[\mathrm{H_{BP1}}=\mathrm{H_{BP2}}=\mathrm{g_{m2}}\mathrm{R_{1}} \tag{15}\]  
Figure 1:0  
From Equations (15) and (16), it is important to note that the passband gains of the NBPF responses are independently adjustable by \(\mathrm{g_{m2}}\) without affecting the NBPF parameters of \(\mathrm{f_{o}}\) and Q. Moreover, the resistors \(\mathrm{R_{2}}\) or \(\mathrm{R_{3}}\) in Equation (16) can achieve a wide range for the independent amplitude adjustable passband gain of the NBPF without affecting parameters \(\mathrm{f_{o}}\) and Q. It is interesting to note that, as shown in Figure 2a, by feeding back the input signal to the output response of the NBPF, a VM fully uncoupled QSO can be realized as shown in Figure 2b. This means that the input signal \(\mathrm{V_{in}}\) is connected to \(\mathrm{V_{BP1}}\) (Figure 2a), and Equation (10) becomes  
Figure 1:1  
Therefore, the CE of Figure 2b is expressed as  
Figure 1:2  
Based on Equation (18), the CO and FO of the proposed VM QSO (Figure 2b) are given by Equations (19) and (20), respectively:  
Figure 1:3  
Figure 1:4  
According to Equations (19) and (20), the CO can be fully controlled independently without affecting the FO by adjusting \(\mathrm{g_{m2}}\) or \(\mathrm{R_{1}}\), and the FO can be also fully controlled independently without affecting the CO by adjusting \(\mathrm{g_{m1}}\) or \(\mathrm{g_{m3}}\). Hence, the VM QSO (Figure 2b) provides a fully uncoupled tuning law for the CO and FO.