Basic characteristics

Figure 1: (a)Active inductor using a gyrator (b)Equivalent circuit of (a)

One of the most popular ways to realize an active inductor is using a gyrator. Figure 1(a) shows an active inductor based on a gyrator. A gyrator consists of two transconductors (OTAs) whose input terminals are connected to an output terminal of another transconductor. When a capacitor $C$ is connected to one of input terminals, input admittance at another input terminal becomes

$$Y_{in}=\frac{i_{in}}{v_{in}}=\frac{g_{m1}g_{m2}}{sC}$$ (1)

where $g_{m1}$ and $g_{m2}$ are transconductance of two transconductors. Equation (1) means that input admittance of Fig. 1(a) and that of Fig. 1(b) are equal.

Figure 2: (a) Another approach to realize active inductor, (b)equivalent circuit of (a)

Figure 2(a) shows another active inductor. A capacitor $C$ is inserted between two input terminals of transconductors. Input admittance of the active inductor is

$$Y_{in}=\frac{i_{in}}{v_{in}}=\frac{g_{m1}g_{m2}}{sC}+(g_{m2}-g_{m1})$$ (2)

and it is equivalent to parallel connection of an inductor and a conductor as shown in Fig. 2(b). Only when $g_{m1}$ and $g_{m2}$ are equal, Fig. 2(a) acts as a pure inductor.

Transconductors used for an active inductors are usually realized by MOSFETs. Actual transconductors have non-ideal characteristics, for example, they have an input capacitance and an output resistance. Taking these parasitic elements into consideration a more accurate equivalent circuit of an active inductor shown in Fig. 3 is obtained. A active inductor contains a series resistance $R_s$, a parallel resistance $R_p$ and input capacitance $C_p$ except inductor $L$. A lot of active inductors can be represented by this equivalent circuit. Values of elements depend on the circuit configuration. Elements values shown in Fig.3 are those of Fig. 1(a) where $C_{i}$ and $r_{o}=1/g_{o}$ are an input capacitance and an output resistance of transconductors.

Figure 3: Equivalent circuit of Fig.1(a)

Input impedance of Fig. 3 depends on a frequency. The frequency range where an active inductor simulates an inductor is

$$f_{l}=\frac{R_s}{2\pi L} \le f \le \frac{1}{2\pi \sqrt{LC_p}}=f_{h} .$$ (3)

The frequency range is depends only on $R_s$ and $C_p$ because usually $R_p$ is much lager than $R_s$. Moreover, even when $R_p$ is comparable to $R_s$, effects of $R_p$ can be cancelled by connecting a negative resistor in parallel with a active inductor.