Ideal gyrator and floating inductor simulation

Ideal gyrator

An ideal gyrator is a lossless two-port that inverts impedance: a load $Z_L$ on port 2 appears at port 1 as $Z_{in} = r^2/Z_L$, where $r$ is the gyration resistance. Its defining equations are

$$V_1 = -r I_2, \qquad V_2 = r I_1 \quad\Longrightarrow\quad Z_{in} = \frac{V_1}{I_1} = \frac{r^2}{Z_L}$$

So a capacitor $C$ on port 2 gives $Z_{in} = r^2 sC = sL$ with $L = r^2C$. A gyrator is built with op-amps (e.g. two voltage-controlled current sources of opposite sign, $I_1 = gV_2$, $I_2 = -gV_1$, $r = 1/g$).

Floating inductor using gyrators

One gyrator with a grounded capacitor gives only a grounded inductor. A floating inductor is made by cascading two identical gyrators with a grounded capacitor between them:

        +-----------+          +-----------+
 a o----| Gyrator 1 |----+-----| Gyrator 2 |----o b
        |    (r)    |    |     |    (r)    |
        +-----------+   === C  +-----------+
                         |
 GND ---------------------------------------------- GND

Each gyrator and the capacitor share the common ground. The transmission (ABCD) matrix of an ideal gyrator ($V_1 = rI_2$, $I_1 = V_2/r$, with $I_2$ leaving port 2) and of a shunt capacitor are

$$G = \begin{bmatrix} 0 & r \\ 1/r & 0 \end{bmatrix}, \qquad Y_C = \begin{bmatrix} 1 & 0 \\ sC & 1 \end{bmatrix}$$

For the cascade:

$$\begin{aligned} G\,Y_C\,G &= \begin{bmatrix} 0 & r \\ 1/r & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ sC & 1 \end{bmatrix}\begin{bmatrix} 0 & r \\ 1/r & 0 \end{bmatrix} \\ &= \begin{bmatrix} rsC & r \\ 1/r & 0 \end{bmatrix}\begin{bmatrix} 0 & r \\ 1/r & 0 \end{bmatrix} = \begin{bmatrix} 1 & sCr^2 \\ 0 & 1 \end{bmatrix} \end{aligned}$$

The matrix $\begin{bmatrix} 1 & Z \\ 0 & 1 \end{bmatrix}$ is that of a single series impedance $Z$ between the two terminals. Here $Z = s\,r^2C$, so the circuit between a and b is a floating inductor

$$L = r^2 C$$

Example: $r$ = 10 kΩ and $C$ = 10 nF give $L = 10^8 \times 10^{-8}$ = 1 H. The two gyrators must have equal gyration resistance; any mismatch leaves an unwanted grounded element. The capacitor is grounded, which suits IC realisation.