## 2 Grounded Gyrator-C-Based Tunable Active Inductor
### Proposed Grounded Tunable Active Inductor  
The grounded active inductor topology [23; 32] remains a widely adopted approach for emulating inductive behavior in integrated circuits, leveraging the well-established gyrator-C architecture. In this configuration, two active devices are arranged to generate an inductive input impedance, with the overall performance primarily determined by the choice of transconductor types. As shown in Figure 4, the fundamental transconductor configurations include the common-source (CS) structure exhibiting negative transconductance, and the common-drain (CD) and common-gate (CG) structures, which provide positive transconductance. To address the limitations of conventional active inductor topologies, namely, low inductance values and a narrow frequency range for achieving high quality factors \(\mathrm{Q}\), this work employs an enhanced design strategy that integrates cascode and feedback-based techniques. Cascode and regulated cascode configurations are introduced to reduce the output conductance of the transistors, thereby decreasing the equivalent series resistance \(\mathrm{R_{s}}\) and improving the quality factor [33]. To further enhance performance, a feedback resistor \(\mathrm{R_{f}}\) is incorporated between the transconductance stages. This feedback path concurrently increases the equivalent inductance \(\mathrm{L_{eq}}\) and reduces \(\mathrm{R_{s}}\), thus enhancing the overall \(\mathrm{Q}\) of the circuit [34]. As \(\mathrm{R_{f}}\) also affects the self-resonant frequency and the frequency at which the maximum quality factor is achieved, i.e., \(f(\mathrm{Q_{max}})\), making \(\mathrm{R_{f}}\) tunable providesindependent control of L, Q, and \(f(\text{Q}_{\text{max}})\), a key requirement for realizing a fully tunable active inductor (TAI). The active inductor proposed in this work, along with its corresponding small-signal equivalent circuit, is depicted in Figure 5a,b. It consists of a single-port grounded structure, formed by two transconductors connected in a back-to-back configuration. Transistor \(\text{M}_{1}\) functions as a transconductor with negative transconductance in a common-source topology, while transistor \(\text{M}_{2}\) provides positive transconductance in a common-drain topology. All transistors operate in the saturation region to ensure proper biasing and linear operation. A cascode transistor \(\text{M}_{3}\) is employed in the feedback path to enhance the quality factor Q by increasing the DC gain, resulting in a reduction in the series resistance \(\text{R}_{\text{s}}\). This reduction leads to a lower zero frequency, thereby broadening the effective inductive frequency range. In addition, transistor \(\text{M}_{4}\), operating in the linear region, functions as a feedback resistor \(\text{R}_{\text{f}}\). By decreasing \(\text{R}_{\text{s}}\) via the feedback resistance, the quality factor is further improved. The resistance can be tuned over a desired range by adjusting the gate voltage \(\text{V}_{\text{ctrl}}\). The circuit shown in Figure 5a requires two current sources to bias transistors \(\text{M}_{1}\), \(\text{M}_{3}\), and \(\text{M}_{2}\). Transistors \(\text{M}_{5}\), \(\text{M}_{6}\), \(\text{M}_{7}\), and \(\text{M}_{8}\) serve as the source supply terminals, where \(\text{M}_{5}\) sources current \(\text{I}_{1}\), and \(\text{M}_{8}\) sinks current and supplies \(\text{I}_{2}\) to the active inductor. Figure 4: Fundamental transconductor configurations: (**a**) common source, (**b**) common drain, and (**c**) common gate. Figure 5: (**a**) Proposed grounded gyrator-C-based active inductor topology and (**b**) the corresponding small signal equivalent circuit. The parallel capacitance \(C_{p}\), parallel resistance \(R_{p}\), series resistance \(R_{s}\), and equivalent inductance \(L_{eq}\) of the active inductor are derived as shown in Equations (9), (10), (11), and (12), respectively.