Conventional active inductor

The circuit configuration of an active inductor depends on that of transconductors. When transconductors is realized by one MOSFET and a bias current source, two kinds of active inductors shown in Fig. 4 and Fig. 5 are obtained.

Figure 4: Active inductor using a source common transconductor and a drain common transconductor

Figure 5: Active inductor using a source common transconductor and a gate common transconductor

The active inductor shown in Fig. 4 consists of a source common transconductor M$_2$ and a drain common transconductor M$_1$. It is obvious that Fig. 4 acts as an active inductor because a configuration of the active inductor is the same with Fig. 2(a). A gate-to-source parasitic capacitance of M$_1$ is utilized for $C$.

On the other hand, transconductors used in Fig. 5 are a source common transconductor M$_2$ and a gate common transconductor M$_1$. Bias current of M$_1$ and M$_2$ is shared to reduce its power consumption. The configuration of Fig. 5 is the same with Fig. 1(a). Gate-to-source capacitances of M$_1$ or M$_2$ is used as $C$.

Input admittance of the active inductors shown in Fig. 4 and Fig. 5 are shown in Eqs. (4) and (5) respectively. Because both of Eqs. (4) and (5) can be represented by

$$Y_{in} = G_{p}+sC_{p}+\frac{1}{sL+R_s},$$ (4)

their equivalent circuits become Fig.3. Values of each element in the equivalent circuit for Fig. 4 and Fig. 5 are

$$G_{p} \simeq g_{m2}~~or~~g_{m1}$$ (5)

$$C_{p} = C_{gs2}~~or~~C_{gs1}$$ (6)

$$R_{s} \simeq \frac{2g_{ds}}{g_{m1}g_{m2}}$$ (7)

$$L \simeq \frac{C_{gs1}}{g_{m1}g_{m2}}~~or~~\frac{C_{gs2}}{g_{m1}g_{m2}}$$ (8)

where $G_p=1/R_p$.

Figure 4 requires about twice bias current compared to Fig. 5. On the other hand, it has wider input voltage range. Self resonance frequency and $Q$ of Fig. 5 are

$$\omega_0 = \frac{1}{\sqrt{C_p L}} = \sqrt{\frac{g_{m1}g_{m2}}{C_{gs1}C_{gs2}}}$$ (9)

$$Q = R_p\omega_0C_p=\sqrt{\frac{g_{m2}C_{gs1}}{g_{m1}C_{gs2}}} .$$ (10)

Enlarging $g_{m2}$ without increasing $g_{m1}$ is effective in maximizing both of $\omega_0$ and $Q$ at the same time. In order to realize this condition a MOSFET M$_4$ is added so that a part of bias current of M$_2$ bypasses M$_1$ as shown in Fig. 6.

Figure 6: Active inductor with high Q