Gyrator simulation of inductor

Ideal gyrator

An ideal gyrator is a lossless two-port that converts a voltage at one port into a proportional current at the other. With gyration conductance $g$ (gyration resistance $r = 1/g$):

$$I_1 = g V_2, \qquad I_2 = -g V_1$$

 I1 ->                    <- I2
 o------+  +---------+  +------o
  V1    |--| gyrator |--|    V2
 o------+  |   g     |  +------o
           +---------+

It is non-reciprocal and inverts impedance: a load $Z_L$ at port 2 appears at port 1 as $Z_{in} = \dfrac{1}{g^2Z_L} = \dfrac{r^2}{Z_L}$.

Simulating an inductor

Terminate port 2 with a capacitor $C$. The load current is $sCV_2$, and it leaves port 2, so $I_2 = -sCV_2$.

$$\begin{aligned} -sCV_2 &= -gV_1 \;\Rightarrow\; V_2 = \frac{g}{sC}V_1 \\ I_1 &= gV_2 = \frac{g^2}{sC}V_1 \\ Z_{in} &= \frac{V_1}{I_1} = s\frac{C}{g^2} = s\,r^2C \end{aligned}$$

So the input looks like an inductor

$$L_{eq} = \frac{C}{g^2} = r^2C$$

 o----+  +---------+  +----+
      |--| gyrator |--|   === C   => L = r^2 C
 o----+  +---------+  +----+

Example: $r = 10\ \text{k}\Omega$, $C = 0.1\ \mu$F gives $L = 10^8 \times 10^{-7} = 10$ H.

Practical realization: a gyrator is built with two voltage-controlled current sources of opposite sign (transconductance $\pm g$), or with op-amps (e.g. the Riordan circuit). Antoniou's GIC can also simulate a grounded inductor directly.

Grounded vs floating: one gyrator with a grounded C gives a grounded inductor. A floating inductor needs two gyrators connected back to back with a grounded capacitor between them, giving $L = r^2C$ between the two outer terminals.

Simulated inductors are small, have high Q and can be integrated, so a passive LC ladder can be converted to an active RC filter while keeping its low sensitivity.