# Gyrator

Last updated

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. [1] 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 conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. [2] 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.

Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems. A passive component, depending on field, may be either a component that consumes but does not produce energy or a component that is incapable of power gain.

A two-port network is an electrical network (circuit) or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the electric current entering one terminal must equal the current emerging from the other terminal on the same port. The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port.

The lumped-element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems, mechanical multibody systems, heat transfer, acoustics, etc.

## Contents

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.

A current–voltage characteristic or I–V curve is a relationship, typically represented as a chart or graph, between the electric current through a circuit, device, or material, and the corresponding voltage, or potential difference across it.

An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

## Behaviour

An ideal gyrator is a linear two port device which couples the current on one port to the voltage on the other and vice versa. The instantaneous currents and instantaneous voltages are related by

${\displaystyle v_{2}=Ri_{1}}$
${\displaystyle v_{1}=-Ri_{2}}$

where ${\displaystyle \scriptstyle {R}}$ 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. [3] 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.

${\displaystyle P=v_{1}i_{1}+v_{2}i_{2}=(-Ri_{2})i_{1}+(Ri_{1})i_{2}\equiv 0}$

A gyrator is an entirely non-reciprocal device, and hence is represented by antisymmetric impedance and admittance matrices:

${\displaystyle Z={\begin{bmatrix}0&-R\\R&0\end{bmatrix}},\quad Y={\begin{bmatrix}0&G\\-G&0\end{bmatrix}},\quad G={\frac {1}{R}}}$
Customary [4]
ANSI Y32 [5] & IEC standards
Two versions of the symbol used to represent a gyrator in single-line diagrams. A 180° (π radian) phase shift occurs for signals travelling in the direction of the arrow (or longer arrow), with no phase shift in the reverse direction.

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

${\displaystyle S={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}}$

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, [6] 180° phase-shifts the backward travelling signal [7] ). 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 of port of the gyrator is terminated with a linear load, then the other port presents an impedance inversely proportional to that of the load,

${\displaystyle \ Z_{\mathrm {in} }={\frac {R^{2}}{Z_{\mathrm {load} }}}}$

A generalization of the gyrator is conceivable, in which the forward and backward gyration conductances have different magnitudes, so that the admittance matrix is

${\displaystyle Y={\begin{bmatrix}0&G_{1}\\-G_{2}&0\end{bmatrix}}}$

However this no longer represents a passive device. [8]

## Name

Tellegen named the element gyrator as a portmanteau 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. [9]

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 vice versa. A mechanical-electrical analogy of the gyroscope making torque and angular velocity the analogs of voltage and current results in the electrical gyrator. [10]

## 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. [1]

Cascaded gyrators of gyration resistance ${\displaystyle \scriptstyle {R_{1}}}$ and ${\displaystyle \scriptstyle {R_{2}}}$ are equivalent to a transformer of turns ratio ${\displaystyle \scriptstyle {R_{1}:R_{2}}}$. Cascading a transformer and a gyrator, or equivalently cascading three gyrators produces a single gyrator of gyration resistance ${\displaystyle \scriptstyle {\frac {R_{1}R_{3}}{R_{2}}}}$.

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 ${\displaystyle r}$ ohms per turn, a loop EMF, ${\displaystyle V}$, is related to a core MMF, ${\displaystyle {\mathcal {F}}}$, by

${\displaystyle V=r{\mathcal {F}}}$

and the loop current ${\displaystyle I}$ is related to the core flux-rate ${\displaystyle {\dot {\Phi }}}$ by

${\displaystyle I={\frac {1}{r}}{\frac {\partial }{\partial t}}\Phi }$

The core of a real, non-ideal, transformer has finite permeance ${\displaystyle {\mathcal {P}}}$ (non-zero reluctance ${\displaystyle {\mathcal {R}}}$), such that the flux and total MMF satisfy

${\displaystyle \Phi ={\frac {\mathcal {F}}{\mathcal {R}}}={\mathcal {P}}{\mathcal {F}}}$

which means that in the gyrator loop

${\displaystyle I={\frac {\mathcal {P}}{r^{2}}}{\frac {\partial }{\partial t}}V}$

corresponding to the introduction of a series capacitor

${\displaystyle C={\frac {1}{r^{2}}}{\mathcal {P}}}$

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:

${\displaystyle Z=R_{\mathrm {L} }+j\omega L\,\!}$

From the diagram, the input impedance of the op-amp circuit is:

${\displaystyle Z_{\mathrm {in} }=\left(R_{\mathrm {L} }+j\omega R_{\mathrm {L} }RC\right)\|\left(R+{1 \over {j\omega C}}\right)}$

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 such that the first term dominates; thus, the RC circuit's effect on input impedance is negligible.

${\displaystyle Z_{\mathrm {in} }\approx R_{\mathrm {L} }+j\omega R_{\mathrm {L} }RC\,\!}$

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.

Grounding. The fact that one side of the simulated inductor is grounded restricts the possible applications (real inductors are floating). This limitation may preclude its use in some low-pass and notch filters. [11] However the gyrator can be used in a floating configuration with another gyrator so long as the floating "grounds" are tied together. This allows for a floating gyrator, but the inductance simulated across the input terminals of the gyrator pair must be cut in half for each gyrator to ensure that the desired inductance is met (the impedance of inductors in series adds together). This is not typically done as it requires even more components than in a standard configuration and the resulting inductance is a result of two simulated inductors, each with half of the desired inductance.

#### Applications

The primary application for a gyrator is to reduce the size and cost of a system by removing the need for bulky, heavy and expensive inductors. For example, RLC bandpass filter characteristics can be realized with capacitors, resistors and operational amplifiers without using inductors. Thus graphic equalizers can be achieved with capacitors, resistors and operational amplifiers without using inductors because of the invention of the gyrator.

Gyrator circuits are extensively used in telephony devices that connect to a POTS system. This has allowed telephones to be much smaller, as the gyrator circuit carries the DC part of the line loop current, allowing the transformer carrying the AC voice signal to be much smaller due to the elimination of DC current through it. [12] Gyrators are used in most DAAs (data access arrangements). [13] Circuitry in telephone exchanges has also been affected with gyrators being used in line cards. Gyrators are also widely used in hi-fi for graphic equalizers, parametric equalizers, discrete bandstop and bandpass filters such as rumble filters), and FM pilot tone filters.

There are many applications where it is not possible to use a gyrator to replace an inductor:

• High voltage systems utilizing flyback (beyond working voltage of transistors/amplifiers)
• RF systems commonly use real inductors as they are quite small at these frequencies and integrated circuits to build an active gyrator are either expensive or non-existent. However, passive gyrators are possible.
• Power conversion, where a coil is used as energy storage.

## Passive gyrators

Numerous passive circuits exist in theory for a gyrator function. However, when constructed of lumped elements there are always negative elements present. These negative elements have no corresponding real component so cannot be implemented in isolation. Such circuits can be used in practice, in filter design for instance, if the negative elements are absorbed into an adjacent positive element. Once active components are permitted, however, a negative element can easily be implemented with a negative impedance converter. For instance, a real capacitor can be transformed into an equivalent negative inductor.

In microwave circuits, impedance inversion can be achieved using a quarter-wave impedance transformer instead of a gyrator. The quarter-wave transformer is a passive device and is far simpler to build than a gyrator. Unlike the gyrator, the transformer is a reciprocal component. The transformer is an example of a distributed-element circuit. [14]

## In other energy domains

Analogs of the gyrator exist in other energy domains. The analogy with the mechanical gyroscope has already been pointed out in the name section. Also, when systems involving multiple energy domains are being analysed as a unified system through analogies, such as mechanical-electrical analogies, the transducers between domains are considered either transformers or gyrators depending on which variables they are translating. [15] Electromagnetic transducers translate current into force and velocity into voltage. In the impedance analogy however, force is the analog of voltage and velocity is the analog of current, thus electromagnetic transducers are gyrators in this analogy. On the other hand, piezoelectric transducers are transformers (in the same analogy). [16]

Thus another possible way to make an electrical passive gyrator is to use transducers to translate into the mechanical domain and back again, much as is done with mechanical filters. Such a gyrator can be made with a single mechanical element by using a multiferroic material using its magnetoelectric effect. For instance, a current carrying coil wound around a multiferroic material will cause vibration through the multiferroic's magnetostrictive property. This vibration will induce a voltage between electrodes embedded in the material through the multiferroic's piezoelectric property. The overall effect is to translate a current into a voltage resulting in gyrator action. [17] [18] [19] [20]

## Related Research Articles

A rectifier is an electrical device that converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in only one direction.

Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped-element circuit model. In other cases infinitesimal elements are used to model the network, in a distributed-element model.

A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

In electronics, impedance matching is the practice of designing the input impedance of an electrical load or the output impedance of its corresponding signal source to maximize the power transfer or minimize signal reflection from the load.

In electronics, a voltage divider is a passive linear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin). Voltage division is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

Practical capacitors and inductors as used in electric circuits are not ideal components with only capacitance or inductance. However, they can be treated, to a very good degree of approximation, as being ideal capacitors and inductors in series with a resistance; this resistance is defined as the equivalent series resistance (ESR). If not otherwise specified, the ESR is always an AC resistance, which means it is measured at specified frequencies, 100 kHz for switched-mode power supply components, 120 Hz for linear power-supply components, and at its self-resonant frequency for general-application components. Additionally, audio components may report a "Q factor", incorporating ESR among other things, at 1000 Hz.

A network, in the context of electronics, is a collection of interconnected components. Network analysis is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values. However, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to linear network analysis.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

The negative impedance converter (NIC) is a one-port op-amp circuit acting as a negative load which injects energy into circuits in contrast to an ordinary load that consumes energy from them. This is achieved by adding or subtracting excessive varying voltage in series to the voltage drop across an equivalent positive impedance. This reverses the voltage polarity or the current direction of the port and introduces a phase shift of 180° (inversion) between the voltage and the current for any signal generator. The two versions obtained are accordingly a negative impedance converter with voltage inversion (VNIC) and a negative impedance converter with current inversion (INIC). The basic circuit of an INIC and its analysis is shown below.

Ripple in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.

The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic fields, similar to magnetic circuits, but based on using elements analogous to capacitors rather than elements analogous to resistors to represent the magnetic flux path. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

The impedance analogy is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

Mechanical–electrical analogies are the representation of mechanical systems as electrical networks. At first, such analogies were used in reverse to help explain electrical phenomena in familiar mechanical terms. James Clerk Maxwell introduced analogies of this sort in the 19th century. However, as electrical network analysis matured it was found that certain mechanical problems could more easily be solved through an electrical analogy. Theoretical developments in the electrical domain that were particularly useful were the representation of an electrical network as an abstract topological diagram using the lumped element model and the ability of network analysis to synthesise a network to meet a prescribed frequency function.

The mobility analogy, also called admittance analogy or Firestone analogy, is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

A frequency dependent negative resistor (FDNR) is a circuit element that exhibits a purely real negative resistance -1/(ω2kC) that decreases in magnitude at a rate of -40 dB per decade. The element is used in implementation of low-pass active filters modeled from ladder filters. The element is usually implemented from a generalized impedance converter (GIC) or gyrator. The impedance of a FDNR is

## References

1. B. D. H. Tellegen (April 1948). "The gyrator, a new electric network element" (PDF). Philips Res. Rep. 3: 81–101. Archived from the original on 2014-04-23. Retrieved 2010-03-20.CS1 maint: BOT: original-url status unknown (link)
2. K. M. Adams, E. F. A. Deprettere and J. O. Voorman (1975). Ladislaus Marton (ed.). "The gyrator in electronic systems". Advances in Electronics and Electron Physics. Academic Press, Inc. 37: 79–180. doi:10.1016/s0065-2539(08)60537-5. ISBN   9780120145379.
3. Chua, Leon, EECS-100 Op Amp Gyrator Circuit Synthesis and Applications (PDF), Univ. of Calif. at Berkeley, retrieved May 3, 2010
4. Fox, A. G.; Miller, S. E.; Weiss, M. T.. (January 1955). "Behavior and Applications of Ferrites in the Microwave Region" (PDF). The Bell System Technical Journal . 34 (1): 5–103. doi:10.1002/j.1538-7305.1955.tb03763.x.
5. Graphic Symbols for Electrical and Electronics Diagrams (Including Reference Designation Letters): IEEE-315-1975 (Reaffirmed 1993), ANSI Y32.2-1975 (Reaffirmed 1989), CSA Z99-1975. IEEE and ANSI, New York, NY. 1993.
6. The IEEE Standard Dictionary of Electrical and Electronics terms (6th ed.). IEEE. 1996 [1941]. ISBN   1-55937-833-6.
7. Theodore Deliyannis, Yichuang Sun, J. Kel Fidler, Continuous-time active filter design, pp.81-82, CRC Press, 1999 ISBN   0-8493-2573-0.
8. Arthur Garratt, "Milestones in electronics: an interview with professor Bernard Tellegen", Wireless World, vol. 85, no. 1521, pp. 133-140, May 1979.
9. Forbes T. Brown, Engineering System Dynamics, pp. 56-57, CRC Press, 2006 ISBN   0849396484.
10. Carter, Bruce (July 2001). "An audio circuit collection, Part 3" (PDF). Analog Applications Journal. Texas Instruments. SLYT134.. Carter page 1 states, "The fact that one side of the inductor is grounded precludes its use in low-pass and notch filters, leaving high-pass and band-pass filters as the only possible applications."
11. Matthaei, George L.; Young, Leo and Jones, E. M. T. Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp. 434-440, McGraw-Hill 1964 (1980 edition is ISBN   0-89006-099-1).
12. Clarence W. de Silva, Mechatronics: An Integrated Approach, pp. 62-65, CRC Press, 2004 ISBN   0203502787.
13. Forbes T. Brown, Engineering System Dynamics, pp. 57-58, CRC Press, 2006 ISBN   0849396484.
14. M. I. Bichurin, V. M. Petrov and S.Priya, "Magnetoelectric multiferroic composites", sect. 8.2 "Magnetoelecric gyrators", chapt. 12 in, Mickaël Lallart (ed), Ferroelectrics - Physical Effects, Intech, 2011 online, ISBN   978-953-307-453-5
15. Haribabu Palneedi, Venkateswarlu Annapureddy, Shashank Priya and Jungho Ryu, "Status and perspectives of multiferroic magnetoelectric composite materials and applications", Actuators, vol. 5, iss. 1, sect. 5, 2016.
16. Nian X. Sun and Gopalan Srinivasan, "Voltsage control of magnetism in multiferroic heterostructures and devices", Spin, vol.2, 2012, 1240004.
17. Junyi Zhai, Jiefang Li, Shuxiang Dong, D. Viehland, and M. I. Bichurin, "A quasi(unidirectional) Tellegen gyrator", J. Appl. Phys., vol.100, 2006, 124509.
• Berndt, D. F.; Dutta Roy, S. C. (1969), "Inductor simulation with a single unity gain amplifier", IEEE Journal of Solid-State Circuits, SC-4: 161–162, doi:10.1109/JSSC.1969.1049979