A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the four conventional elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.
Tellegen invented a circuit symbol for the gyrator and suggested a number of ways in which a practical gyrator might be built. An important property of a gyrator is that it inverts the current–voltage characteristic of an electrical component or network. In the case of linear elements, the impedance is also inverted. In other words, a gyrator can make a capacitive circuit behave inductively, a series LC circuit behave like a parallel LC circuit, and so on. It is primarily used in active filter design and miniaturization.
Behavior
An ideal gyrator is a linear 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
v 2 = R i 1 , {\displaystyle v_{2}=Ri_{1},}
v 1 = − R i 2 , {\displaystyle v_{1}=-Ri_{2},}
where R {\displaystyle R} is the gyration resistance of the gyrator. The gyration resistance (or equivalently its reciprocal the gyration conductance) has an associated direction indicated by an arrow on the schematic diagram. 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:
P = v 1 i 1 + v 2 i 2 = ( − R i 2 ) i 1 + ( R i 1 ) i 2 ≡ 0. {\displaystyle P=v_{1}i_{1}+v_{2}i_{2}=(-Ri_{2})i_{1}+(Ri_{1})i_{2}\equiv 0.}
A gyrator is an entirely non-reciprocal device, and hence is represented by antisymmetric impedance and admittance matrices:
Z = [ 0 − R R 0 ] , Y = [ 0 G − G 0 ] , G = 1 R . {\displaystyle Z={\begin{bmatrix}0&-R\\R&0\end{bmatrix}},\quad Y={\begin{bmatrix}0&G\\-G&0\end{bmatrix}},\quad G={\frac {1}{R}}.}
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 matrix for the gyrator is
S = [ 0 − 1 1 0 ] , {\displaystyle S={\begin{bmatrix}0&-1\\1&0\end{bmatrix}},}
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, 180° phase-shifts the backward-travelling signal). The symbol used to represent a gyrator in one-line diagrams (where a 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 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:
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