Gyrators#

Gyrators are electronic components which are capable of emulating duals.

Theoretical Model#

Definition: Gyrator

A gyrator is a time-invariant two-port for which there exists some \(R_g \in \mathbb{R}\) or \(G_g \in \mathbb{R}\) with the following property:

\[ \left\vert\begin{aligned}v_1 &= -R_g i_2 \\ v_2 &= R_g i_1\end{aligned}\right. \qquad \text{or} \qquad \left\vert\begin{aligned}i_1 &= G_g v_2 \\ i_2 &= -G_g v_1\end{aligned}\right. \]

We call \(R_g\) the gyration resistance and \(G_g\) the gyration conductance.

Notation

The following symbol is used for the gyrator:

Gyrator Symbol

The arrow is optional. When it is present, the convention is that it connects the current at its tail to the voltage at its head.

Theorem: Strict Linearity

Every gyrator is strictly linear.

Proof

TODO

Theorem: Reciprocity of \(R_g\) and \(G_g\)

If a gyrator has both a gyration resistance \(R_g\) and a gyration conductance \(G_g\), then they are reciprocal:

\[ R_g = \frac{1}{G_g} \qquad G_g = \frac{1}{R_g} \]

Proof

TODO

Theorem: Implicit Representations

If a gyrator has gyration resistance \(R_g\), then it has the following implicit representation:

\[ \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\boldsymbol{v} + \begin{bmatrix}0 & R_g \\ -R_g & 0\end{bmatrix}\boldsymbol{i} = \boldsymbol{0} \]

If a gyrator has gyration conductance \(G\), then it has the following implicit representation:

\[ \begin{bmatrix}0 & -G_g \\ G_g & 0\end{bmatrix}\boldsymbol{v} + \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\boldsymbol{i} = \boldsymbol{0} \]

Proof

TODO

Theorem: Explicit Representations

If a gyrator has gyration conductance \(G_g\), then it has the following admittance representation:

\[ \boldsymbol{i} = \boldsymbol{G}\boldsymbol{v} \qquad \boldsymbol{G} = \begin{bmatrix}0 & G_g \\ -G_g & 0\end{bmatrix} \]

If a gyrator has gyration resistance \(R_g\), then it has the following impedance representation:

\[ \boldsymbol{v} = \boldsymbol{R}\boldsymbol{i} \qquad \boldsymbol{R} = \begin{bmatrix}0 & -R_g \\ R_g & 0\end{bmatrix} \]

Gyrators have neither hybrid representations nor inverse hybrid representations.

If a gyrator has both gyration conductance \(G_g\) and gyration resistance \(R_g\), then it has the following forwards transmission representation:

\[ \begin{bmatrix}v_1 \\ i_1\end{bmatrix} = \boldsymbol{T}\begin{bmatrix}v_2 \\ -i_2\end{bmatrix} \qquad \boldsymbol{T} = \begin{bmatrix}0 & R_g \\ G_g & 0\end{bmatrix} \]

If a gyrator has both gyration conductance \(G_g\) and gyration resistance \(R_g\), then it has the following backwards transmission representation:

\[ \begin{bmatrix}v_2 \\ i_2\end{bmatrix} = \boldsymbol{T}'\begin{bmatrix}v_1 \\ -i_1\end{bmatrix} \qquad \boldsymbol{T}' = \begin{bmatrix}0 & -R_g \\ -G_g & 0\end{bmatrix} \]

Proof

TODO

Theorem: Losslessness of the Gyrator

Gyrators are lossless.

Proof

TODO

Theorem: Positive Immitance Inverter

The gyrator is a positive immitance inverter (PII).

Proof

TODO

Theorem: Dual of the Gyrator

A gyrator with gyration resistance is dual to a gyrator.

Proof

TODO

Theorem: Antisymmetry of Gyrators

Gyrators are antisymmetrical.

Proof

TODO

Theorem: Gyrator as One-Port Dual Converter

If the output of a gyrator with gyration resistance \(R_g\) is connected to a one-port \(\mathcal{F}\), then its input becomes the dual of \(\mathcal{F}\) with duality constant \(R_g\):

Gyrator as One-Port Dual Converter

Proof

TODO

Theorem: Gyrator as Two-Port Dual Converter

The chaining of a two-port \(\mathcal{F}\) with gyrators with gyration resistance \(R_g\) on its input and output, results in its dual two-port \(\mathcal{F}^d\) with duality constant \(R_g\):

Gyrator as Two-Port Dual Converter

Proof

TODO