## 2 Grounded Gyrator-C-Based Tunable Active Inductor
### Proposed Grounded Tunable Active Inductor
For simplification, certain parasitic elements such as the impedance output of transistors and the gate-drain capacitance \(C_{gd}\) are neglected. \[C_{p}=C_{gs1} \tag{9}\]  
\[R_{p}=\frac{1+g_{ds3}R_{f}}{g_{ds3}(2+g_{ds3}R_{f})} \tag{10}\]  
\[R_{s}=\frac{g_{ds1}g_{ds3}g_{m2}+\omega^{2}\Big{(}C_{gs2}^{2}g_{m3}-C_{gs2}C_{gs3 }g_{m2}(1+g_{ds3}R_{f})\Big{)}}{g_{m1}g_{m3}g_{m2}^{2}+\omega^{2}C_{gs2}^{2}g_{ m1}g_{m3}} \tag{11}\]  
\[L_{eq}=\frac{C_{gs2}g_{m2}g_{m3}+\omega^{2}C_{gs2}^{2}C_{gs3}(1+g_{ds3}R_{f})} {g_{m1}g_{m3}g_{m2}^{2}+\omega^{2}C_{gs2}^{2}g_{m1}g_{m3}} \tag{12}\]  
Assuming \(g_{m2}^{2}>>\omega^{2}C_{gs2}^{2}\), the equivalent inductance \(L_{eq}\) can be approximated by:  
\[L_{eq}=\frac{C_{gs2}g_{m2}g_{m3}+\omega^{2}C_{gs2}^{2}C_{gs3}(1+g_{ds3}R_{f})} {g_{m1}g_{m3}g_{m2}^{2}} \tag{13}\]  
where \(g_{ds1}\), \(g_{ds2}\), and \(g_{ds3}\) denote the output conductances; \(C_{gs1}\), \(C_{gs2}\), and \(C_{gs3}\) represent the gate-source capacitances at nodes 1, 2, and 3, respectively; and \(g_{m1}\), \(g_{m2}\), and \(g_{m3}\) are the transconductances of transistors M\({}_{1}\), M\({}_{2}\), and M\({}_{3}\), respectively. From Equation (13), it is evident that the equivalent inductance exhibits an inverse relationship with the transconductance \(g_{m3}\), which is determined by the current flowing through M\({}_{3}\). In this design, the cascode stage, implemented by transistor M\({}_{3}\), is used to control the inductance. Lowering the gate voltage of M\({}_{3}\) increases its drain current, which also impacts the drain current sourced by M\({}_{5}\). Given that transconductance is defined as \(g_{m}=\partial I_{D}/\partial V_{GS}=\sqrt{2\mu_{n}C_{on}\frac{W}{L}}I_{D}\), an increase in drain current results in a higher \(g_{m3}\), thereby leading to a reduction in \(L_{eq}\). The quality factor of the active inductor is given by:  
\[Q=\frac{C_{gs2}g_{m2}g_{m3}\omega+\omega^{3}C_{gs2}^{2}C_{gs3}(Rg_{ds3}+1)}{g _{ds1}g_{ds3}g_{m2}+\omega^{2}(C_{gs2}^{2}g_{m3}-C_{gs2}C_{gs3}g_{m2}(R_{f}g_{ ds3}+1))} \tag{14}\]  
The zero frequency \(\omega_{z}\) and the self-resonant frequency \(\omega_{0}\), which establish the operational frequency boundaries, are expressed as:  
\[\omega_{z}=\frac{g_{ds1}g_{ds3}g_{m2}+\omega^{2}\Big{(}C_{gs2}^{2}g_{m3}-C_{gs 2}C_{gs3}g_{m2}(1+g_{ds3}R_{f})\Big{)}}{C_{gs2}g_{m2}g_{m3}+\omega^{2}\Big{(} C_{gs2}^{2}C_{gs3}(1+g_{ds3}R_{f})\Big{)}} \tag{15}\]  
\[\omega_{0}=\sqrt{\frac{g_{m1}g_{m3}g_{m2}^{2}+\omega^{2}C_{gs2}^{2}g_{m1}g_{m3 }}{C_{gs1}C_{gs2}g_{m2}g_{m3}+\omega^{2}\Big{(}C_{gs2}^{2}C_{gs1}C_{gs3}(1+g_{ ds3}R_{f})\Big{)}}} \tag{16}\]  
Since the additional inductive reactance introduced by the feedback resistor \(R_{f}\) appears in the expressions for both the quality factor Q and the equivalent inductance \(L_{eq}\), its influence varies as a function of this resistance. As indicated by Equation (14), and from the associated Equations (11) and (12), the inclusion of a cascode stage contributes to a reduction in the equivalent series resistance, as \(g_{m3}<1\), thereby improving the quality factor. Moreover, the incorporation of a feedback resistor \(\mathrm{R_{f}}\) significantly enhances Q by simultaneously increasing \(\mathrm{L_{eq}}\) and reducing \(\mathrm{R_{s}}\). By tuning \(\mathrm{R_{f}}\), it is possible to adjust both the inductance and the quality factor. However, as shown by Equation (17) and illustrated in Figure 6, a substantial increase in \(\mathrm{R_{f}}\) results in a pronounced reduction \(\mathrm{inf}_{\mathrm{Q}}\), the frequency at which the quality factor reaches its maximum. To mitigate this limitation, a secondary tuning mechanism is introduced via variation in the output conductance \(\mathrm{g_{ds5}}\), which is controlled using the voltage \(\mathrm{V_{ctr2}}\). Consequently, simultaneous tuning of \(\mathrm{V_{ctr1}}\) and \(\mathrm{V_{ctr2}}\) enables the realization of the desired inductance and high-Q performance at the target operating frequency \(f_{\mathrm{Q}}\).