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
where 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:
A gyrator is an entirely non-reciprocal device, and hence is represented by antisymmetric impedance and admittance matrices:
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
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:
A generalization of the gyrator is conceivable, in which the forward and backward gyration conductances have different magnitudes, so that the admittance matrix is
However, this no longer represents a passive device.Theodore Deliyannis, Yichuang Sun, J. Kel Fidler, Continuous-time active filter design, pp. 81–82, CRC Press, 1999, .
Name Tellegen named the element gyrator as a blend of gyroscope and the common device suffix -tor (as in resistor, capacitor, transistor etc.) The -tor ending is even more suggestive in Tellegen's native Dutch, where the related element transformer is called transformator. The gyrator is related to the gyroscope by an analogy in its behaviour.Arthur Garratt, "Milestones in electronics: an interview with professor Bernard Tellegen", Wireless World, vol. 85, no. 1521, pp. 133–140, May 1979.
The analogy with the gyroscope is due to the relationship between the torque and angular velocity of the gyroscope on the two axes of rotation. A torque on one axis will produce a proportional change in angular velocity on the other axis and conversely. A mechanical–electrical analogy of the gyroscope making torque and angular velocity the analogs of voltage and current results in the electrical gyrator.Forbes T. Brown, Engineering System Dynamics, pp. 56–57, CRC Press, 2006, .
Relationship to the ideal transformer
An ideal gyrator is similar to an ideal transformer in being a linear, lossless, passive, memoryless two-port device. However, whereas a transformer couples the voltage on port 1 to the voltage on port 2, and the current on port 1 to the current on port 2, the gyrator cross-couples voltage to current and current to voltage. Cascading two gyrators achieves a voltage-to-voltage coupling identical to that of an ideal transformer.
Cascaded gyrators of gyration resistance and are equivalent to a transformer of turns ratio . Cascading a transformer and a gyrator, or equivalently cascading three gyrators produces a single gyrator of gyration resistance .
From the point of view of network theory, transformers are redundant when gyrators are available. Anything that can be built from resistors, capacitors, inductors, transformers and gyrators, can also be built using just resistors, gyrators and inductors (or capacitors).
Magnetic circuit analogy
In the two-gyrator equivalent circuit for a transformer, described above, the gyrators may be identified with the transformer windings, and the loop connecting the gyrators with the transformer magnetic core. The electric current around the loop then corresponds to the rate-of-change of magnetic flux through the core, and the electromotive force (EMF) in the loop due to each gyrator corresponds to the magnetomotive force (MMF) in the core due to each winding.
The gyration resistances are in the same ratio as the winding turn-counts, but collectively of no particular magnitude. So, choosing an arbitrary conversion factor of ohms per turn, a loop EMF is related to a core MMF by
and the loop current is related to the core flux-rate by
The core of a real, non-ideal, transformer has finite permeance (non-zero reluctance ), such that the flux and total MMF satisfy
which means that in the gyrator loop
corresponding to the introduction of a series capacitor
in the loop. This is Buntenbach's capacitance–permeance analogy, or the gyrator–capacitor model of magnetic circuits.
Application Simulated inductor
A gyrator can be used to transform a load capacitance into an inductance. At low frequencies and low powers, the behaviour of the gyrator can be reproduced by a small op-amp circuit. This supplies a means of providing an inductive element in a small electronic circuit or integrated circuit. Before the invention of the transistor, coils of wire with large inductance might be used in electronic filters. An inductor can be replaced by a much smaller assembly containing a capacitor, operational amplifiers or transistors, and resistors. This is especially useful in integrated circuit technology.
Operation
In the circuit shown, one port of the gyrator is between the input terminal and ground, while the other port is terminated with the capacitor. The circuit works by inverting and multiplying the effect of the capacitor in an RC differentiating circuit, where the voltage across the resistor R behaves through time in the same manner as the voltage across an inductor. The op-amp follower buffers this voltage and applies it back to the input through the resistor RL. The desired effect is an impedance of the form of an ideal inductor L with a series resistance RL:
From the diagram, the input impedance of the op-amp circuit is
With RLRC = L, it can be seen that the impedance of the simulated inductor is the desired impedance in parallel with the impedance of the RC circuit. In typical designs, R is chosen to be sufficiently large while such that the first term dominates; thus, the RC-circuit's effect on input impedance is negligible:
This is the same as a resistance RL in series with an inductance L = RLRC. There is a practical limit on the minimum value that RL can take, determined by the current output capability of the op-amp.
The impedance cannot increase indefinitely with frequency, and eventually the second term limits the impedance to the value of R.
Comparison with actual inductors
Simulated elements are electronic circuits that imitate actual elements. Simulated elements cannot replace physical inductors in all the possible applications as they do not possess all the unique properties of physical inductors.
Magnitudes. In typical applications, both the inductance and the resistance of the gyrator are much greater than that of a physical inductor. Gyrators can be used to create inductors from the microhenry range up to the megahenry range. Physical inductors are typically limited to tens of henries, and have parasitic series resistances from hundreds of microhms through the low kilohm range. The parasitic resistance of a gyrator depends on the topology, but with the topology shown, series resistances will typically range from tens of ohms through hundreds of kilohms.
Quality. Physical capacitors are often much closer to "ideal capacitors" than physical inductors are to "ideal inductors". Because of this, a synthesized inductor realized with a gyrator and a capacitor may, for certain applications, be closer to an "ideal inductor" than any (practical) physical inductor can be. Thus, use of capacitors and gyrators may improve the quality of filter networks that would otherwise be built using inductors. Also, the Q factor of a synthesized inductor can be selected with ease. The Q of an LC filter can be either lower or higher than that of an actual LC filter – for the same frequency, the inductance is much higher, the capacitance much lower, but the resistance also higher. Gyrator inductors typically have higher accuracy than physical inductors, due to the lower cost of precision capacitors than inductors.
Energy storage. Simulated inductors do not have the inherent energy storing properties of the real inductors and this limits the possible power applications. The circuit cannot respond like a real inductor to sudden input changes (it does not produce a high-voltage back EMF); its voltage response is limited by the power supply. Since gyrators use active circuits, they only function as a gyrator within the power supply range of the active element. Hence gyrators are usually not very useful for situations requiring simulation of the 'flyback' property of inductors, where a large voltage spike is caused when current is interrupted. A gyrator's transient response is limited by the bandwidth of the active device in the circuit and by the power supply.
Externalities. Simulated inductors do not react to external magnetic fields and permeable materials the same way that real inductors do. They also don't create magnetic fields (and induce currents in external conductors) the same way that real inductors do. This limits their use in applications such as sensors, detectors and transducers.