## 2 Grounded Gyrator-C-Based Tunable Active Inductor  
### Basic Concept  
A gyrator can be realized by connecting two transconductors in series, where one exhibits positive transconductance and the other negative transconductance. When a capacitor is connected at the output, the resulting configuration forms a gyrator-C network, as illustrated in Figure 1a. By applying Kirchhoff's Current Law (KCL) at nodes 1 and 2, the input admittance \(\mathrm{Y_{in}}\) can be derived:  
\[\mathrm{Y_{in}}=\frac{\mathrm{I_{in}}}{\mathrm{V_{2}}}=\mathrm{s\,C_{2}}+ \mathrm{G_{o2}}+\frac{1}{\mathrm{s\,C_{1}}}+\frac{\mathrm{G_{o1}}}{\mathrm{G_ {m1}G_{m2}}} \tag{1}\]  
This expression reveals that the gyrator-C structure emulates the behavior of an equivalent RLC network, as represented in Figure 1b, with the corresponding parameters given by:  
\[\mathrm{C_{P}}=\mathrm{C_{2}},\;\mathrm{R_{P}}=\frac{1}{\mathrm{G_{o2}}},\; \mathrm{L_{eq}}=\frac{\mathrm{C_{1}}}{\mathrm{G_{m1}G_{m2}}},\;\mathrm{and}\; \mathrm{R_{s}}=\frac{\mathrm{G_{o1}}}{\mathrm{G_{m1}G_{m2}}} \tag{2}\]  
where \(\mathrm{G_{o1}}\) and \(\mathrm{G_{o2}}\) denote the total input and output conductances of the transconductors at ports 1 and 2, respectively. From Equation (2), a gyrator-C network can replicate the behavior of an inductor \(\mathrm{L_{eq}}\) along with its associated parasitic components: parallel resistance \(\mathrm{R_{P}}\), parallel capacitance \(\mathrm{C_{P}}\), and series resistance \(\mathrm{R_{s}}\). The equivalent inductance is directly proportional to the output capacitance \(\mathrm{C_{1}}\) and inversely proportional to the trans conductances \(\mathrm{G_{m1}}\) and \(\mathrm{G_{m2}}\), allowing for tunability through adjustment of these parameters. However, the inductive behavior is confined to a specific frequency range, which can be determined by analyzing the input impedance \(Z_{\mathrm{in}}\), as defined in Equation (3). \[Z_{\mathrm{in}}=\left(\frac{\mathrm{R_{s}}}{\mathrm{L_{eq}C_{P}}}\right)\frac {\mathrm{s\,\frac{Leq}{R_{s}}+1}}{\mathrm{s^{2}+s\,\frac{1}{R_{P}C_{P}}+\frac {\mathrm{R_{s}}}{Leq})+\frac{\mathrm{R_{s}+R_{P}}}{R_{P}LeqC_{P}}}} \tag{3}\]  
The inductive behavior of a lossy gyrator-C network is limited to a frequency band bounded by the zero frequency \(\omega_{x}\) and the self-resonant frequency \(\omega_{0}\). The self-resonant frequency \(\omega_{0}\) is obtained by ensuring that the phase angle of \(Z_{\mathrm{in}}\) is zero, which occurs when  
Figure 1: (**a**) Lossy single-ended gyrator-C network-based AIs, and (**b**) the corresponding equivalent RLC model. the phase angles of the numerator and denominator are equal. By setting \(\mathbf{\triangle}\ Z_{\text{in}}(\text{j}\omega)=0\) and using s = j\(\omega\), the value of \(\omega_{0}\) can be derived as:  
\[\arctan\measuredangle\left(1+\text{s}\frac{\text{L}_{\text{eq}}}{\text{R}_{\text{ s}}}\right)=\arctan\measuredangle\left(\left(\frac{\text{R}_{\text{s}}+\text{R}_{ \text{P}}}{\text{R}_{\text{P}}\text{L}_{\text{eq}}\text{C}_{\text{P}}}-\omega^ {2}\right)+\text{s}\left(\frac{1}{\text{R}_{\text{P}}\text{C}_{\text{P}}}+ \frac{\text{R}_{\text{s}}}{\text{L}_{\text{eq}}}\right)\right) \tag{4}\]  
By solving Equation (4), \(\omega_{0}\) can be expressed as:  
\[\omega_{0}=\sqrt{\frac{1}{\sqrt{\text{L}_{\text{eq}}\text{C}_{\text{P}}}}- \left(\frac{\text{R}_{\text{s}}}{\text{L}_{\text{eq}}}\right)^{2}} \tag{5}\]  
From Equation (3), the input impedance \(\text{Z}_{\text{in}}\) exhibits a zero at the frequency \(\omega_{\text{z}}=\text{R}_{\text{S}}/\text{L}_{\text{eq}}\). Additionally, \(\text{Z}_{\text{in}}\) has a pole corresponding to the resonant frequency given by \(\omega_{\text{P}}=\sqrt{(\text{R}_{\text{s}}+\text{R}_{\text{P}})/\text{R}_{ \text{P}}\text{L}_{\text{eq}}\text{C}_{\text{P}}}\).