Editing Gyrator (section)

== Behaviour ==
[[File:tellegen-gyrator-annotated.svg|thumb|Gyrator schematic labelled]]
An ideal gyrator is a linear [[two-port network|two-port device]] which couples the current on one port to the voltage on the other and conversely. The instantaneous currents and instantaneous voltages are related by

: <math>v_2 = R i_1,</math>

: <math>v_1 = -R i_2,</math>

where <math>R</math> is the ''gyration [[Electrical resistance|resistance]]'' of the gyrator.

The gyration resistance (or equivalently its reciprocal the ''gyration [[Electrical conductance|conductance]]'') has an associated direction indicated by an arrow on the schematic diagram.<ref name="Chua"/> By convention, the given gyration resistance or conductance relates the voltage on the port at the head of the arrow to the current at its tail. The voltage at the tail of the arrow is related to the current at its head by ''minus'' the stated resistance. Reversing the arrow is equivalent to negating the gyration resistance, or to reversing the polarity of either port.

Although a gyrator is characterized by its resistance value, it is a lossless component. From the governing equations, the instantaneous power into the gyrator is identically zero:

: <math>P = v_1 i_1 + v_2 i_2 = (-R i_2) i_1 + (R i_1) i_2 \equiv 0.</math>

A gyrator is an entirely non-reciprocal device, and hence is represented by [[skew-symmetric matrix|antisymmetric]] [[impedance parameters|impedance]] and [[admittance parameters|admittance matrices]]:
:<math>
  Z = \begin{bmatrix} 0 & -R \\  R & 0 \end{bmatrix},\quad
  Y = \begin{bmatrix} 0 &  G \\ -G & 0 \end{bmatrix},\quad G = \frac{1}{R}.
</math>
{{multiple image
 | width = 100
 | footer = Two versions of the symbol used to represent a gyrator in single-line diagrams. A 180° (π&nbsp;radian) phase shift occurs for signals travelling in the direction of the arrow (or longer arrow), with no phase shift in the reverse direction.
 | image1 = Gyrator-single-line-symbol.svg
 | alt1 = Line interrupted by a box containing the letter pi and an arrow
 | caption1 = Customary<ref name="fox-miller-weiss"/>
 | image2 = Gyrator-single-line-symbol-ANSI-IEC.svg
 | alt2 = Line interrupted by a box containing the letter pi and an arrow
 | caption2 = ANSI Y32<ref name="ieee-315-ansi-y32"/> & IEC standards
 }}
If the gyration resistance is chosen to be equal to the [[characteristic impedance]] of the two ports (or to their [[geometric mean]] if these are not the same), then the [[Scattering parameters|scattering matrix]] for the gyrator is
:<math>
  S = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix},
</math>
which is likewise antisymmetric. This leads to an alternative definition of a gyrator: a device which transmits a signal unchanged in the forward (arrow) direction, but reverses the polarity of the signal travelling in the backward direction (or equivalently,<ref name="hogan1952"/> 180° phase-shifts the backward-travelling signal<ref name="IEEEdict6"/>). The symbol used to represent a gyrator in [[one-line diagram]]s (where a [[waveguide (electromagnetism)|waveguide]] or [[transmission line]] is shown as a single line rather than as a pair of conductors), reflects this one-way phase shift.

As with a [[quarter-wave impedance transformer|quarter-wave transformer]], if one port of a gyrator is terminated with a linear load, then the other port presents an impedance inversely proportional to the impedance of that load:

: <math>Z_\text{in} = \frac{R^2}{Z_\text{load}}.</math>

A generalization of the gyrator is conceivable, in which the forward and backward gyration conductances have different magnitudes, so that the admittance matrix is
: <math>Y = \begin{bmatrix} 0 & G_1 \\ -G_2 & 0 \end{bmatrix}.</math>
However, this no longer represents a passive device.<ref>Theodore Deliyannis, Yichuang Sun, J. Kel Fidler, ''Continuous-time active filter design'', pp.&nbsp;81–82, CRC Press, 1999, {{ISBN|0-8493-2573-0}}.</ref>