Gyrator

A gyrator is an impedance inverter. Thus, for example, it converts a capacitance into an inductance and vice versa. The ideal gyrator, as shown in fig. 9.6, is determined by the following equations which introduce two more unknowns in the MNA matrix.

Figure 9.6: ideal gyrator

$$I_{in} = \frac{1}{R}\cdot\left(V_{2} - V_{3}\right) \quad \rightarrow \quad \frac{1}{R}\cdot V_{2} - \frac{1}{R}\cdot V_{3} - I_{in} = 0$$ (9.133)

$$I_{out} = -\frac{1}{R}\cdot\left(V_{1} - V_{4}\right) \quad \rightarrow \quad -\frac{1}{R}\cdot V_{1} + \frac{1}{R}\cdot V_{4} - I_{out} = 0$$ (9.134)

The new unknown variables $I_{out}$ and $I_{in}$ must be considered by the four remaining simple equations.

$$I_{1} = I_{in} \quad I_{2} = I_{out} \quad I_{3} = -I_{out} \quad I_{4} = -I_{in}$$ (9.135)

As can be seen, a gyrator consists of two cross-connected VCCS (section 9.19.1). Hence, its y-parameter matrix is:

$$(\underline{Y}) = \begin{bmatrix}0&\frac{1}{R}&-\frac{1}{R}&0\\ -... ...{R}\\ \frac{1}{R}&0&0&-\frac{1}{R}\\ 0&-\frac{1}{R}&\frac{1}{R}&0 \end{bmatrix}$$ (9.136)

The scattering matrix of an ideal gyrator with the ratio $R$ writes as follows.

$$r = \frac{R}{Z_{ref}} = \frac{1}{G\cdot Z_{ref}}$$ (9.137)

$$S_{11} = S_{22} = S_{33} = S_{44} = \frac{R^2}{4\cdot Z_{ref}^2 + R^2} = \frac{r^2}{r^2+4}$$ (9.138)

$$S_{14} = S_{23} = S_{32} = S_{41} = 1-S_{11}$$ (9.139)

$$S_{12} = -S_{13} = -S_{21} = S_{24} = S_{31} = -S_{34} = -S_{42} = S_{43} = \frac{2\cdot r}{r^2+4}$$ (9.140)