Gyrator
Encyclopedia
A gyrator is a passive
, linear, lossless, two-port
electrical network element
proposed in 1948 by 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 conventional four 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.
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 element
s, 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
.
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
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
).
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,
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.
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)
The gyrator network can be used to transform a load capacitance into an inductance. The primary use of a gyrator is to simulate an inductive
element in a small electronic circuit
or integrated circuit
. Before the invention of the transistor
, coil
s of wire
with large inductance
might be used in electronic filter
s. An inductor can be replaced by a much smaller assembly containing a capacitor
, operational amplifier
s or transistors, and resistor
s. This is especially useful in integrated circuit technology.
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 such that the first term dominates; thus, the RC circuit's impact 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.
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.
Grounding. The fact that one side of the simulated inductor is grounded restricts the possible applications (real inductors are floating). This limitation precludes its use in low-pass and notch filters, leaving high-pass and band-pass filters as the only possible applications.
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. Circuitry in telephone exchanges has also been affected with gyrators being used in line card
s. 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:
Passivity (engineering)
Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems...
, linear, lossless, two-port
Two-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
electrical network element
Lumped element model
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...
proposed in 1948 by Tellegen
Bernard D. H. Tellegen
Bernard D.H. Tellegen was a Dutch electrical engineer and inventor of the penthode and the gyrator...
as a hypothetical fifth linear element
Linear element
In an electric circuit, a linear element is an electrical element with a linear relationship between current and [output[voltage]]. Resistors are the most common example of a linear element; other examples include capacitors, inductors, and transformers....
after the resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
, capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
, inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal
Reciprocity (electromagnetism)
In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints...
. Gyrators permit network realizations
Network synthesis filters
Network synthesis is a method of designing signal processing filters. It has produced several important classes of filter including the Butterworth filter, the Chebyshev filter and the Elliptic filter. It was originally intended to be applied to the design of passive linear analogue filters but...
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
Isolator (microwave)
An isolator is a two-port device that transmits microwave or radio frequency power in one direction only. It is used to shield equipment on its input side, from the effects of conditions on its output side; for example, to prevent a microwave source being detuned by a mismatched...
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.
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
Current-voltage characteristic
A current–voltage characteristic is a relationship, typically represented as a chart or graph, between an electric current and a corresponding voltage, or potential difference.-In electronics:...
of an electrical component or network
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
. In the case of linear element
Linear element
In an electric circuit, a linear element is an electrical element with a linear relationship between current and [output[voltage]]. Resistors are the most common example of a linear element; other examples include capacitors, inductors, and transformers....
s, the impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
is also inverted. In other words, a gyrator can make a capacitive
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
circuit behave inductively
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
, a series LC circuit behave like a parallel LC circuit, and so on. It is primarily used in active filter
Active filter
An active filter is a type of analog electronic filter that uses an amplifier stage. Amplifiers included in a filter design can be used to improve the performance, stability and predictability of a filter. An amplifier prevents the impedance of source or load stages from affecting the...
design and miniaturization
Miniaturization
Miniaturization is the creation of ever-smaller scales for mechanical, optical, and electronic products and devices...
.
Behaviour
An ideal gyrator is a linear two port deviceTwo-port network
A two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
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
where
is the gyration resistanceElectrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
of the gyrator.
The gyration resistance (or equivalently its reciprocal the gyration conductance
Conductance
Conductance may refer to:* Electrical conductance, the ability for electricity to flow a certain path* Fluid conductance, the ability for fluid to transmit through materials* Thermal conductivity, the ability for temperatures to transmit through materials...
) 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
Reciprocity (electromagnetism)
In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints...
device, and hence is represented by antisymmetric
Skew-symmetric matrix
In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
impedance
Impedance parameters
Impedance parameters or Z-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal response of non-linear networks...
and admittance matrices
Admittance parameters
Admittance parameters or Y-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal response of non-linear networks...
:
If the gyration resistance is chosen to be equal to the characteristic impedance
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
of the two ports (or to their geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
if these are not the same), then the scattering matrix
Scattering parameters
Scattering parameters or S-parameters describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals....
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
).
As with a quarter wave transformer
Quarter wave impedance transformer
A quarter-wave impedance transformer, often written as λ/4 impedance transformer, is a component used in electrical engineering consisting of a length of transmission line or waveguide exactly one-quarter of a wavelength long and terminated in some known impedance. The device presents at its...
, 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,
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.
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)
Implementation: a simulated inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
element in a small electronic circuit
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
or integrated circuit
Integrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
. Before the invention of the transistor
Transistor
A transistor is a semiconductor device used to amplify and switch electronic signals and power. It is composed of a semiconductor material with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals changes the current...
, coil
Coil
A coil is a series of loops. A coiled coil is a structure in which the coil itself is in turn also looping.-Electromagnetic coils:An electromagnetic coil is formed when a conductor is wound around a core or form to create an inductor or electromagnet...
s of wire
Wire
A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads and to carry electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Standard sizes are determined by various...
with large inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
might be used in electronic filter
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
s. An inductor can be replaced by a much smaller assembly containing a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
, operational amplifier
Operational amplifier
An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...
s or transistors, and resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...
s. This is especially useful in integrated circuit technology.
Operation
The circuit works by inverting and multiplying the effect of the capacitor in an RC differentiating circuit where the voltage across the resistor 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 such that the first term dominates; thus, the RC circuit's impact 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.
Comparison with actual inductors
Simulated elements only imitate actual elements as in fact they are dynamic voltage sources. They cannot replace them in all the possible applications as they do not possess all their unique properties. So, the simulated inductor only mimics some properties of the real inductor.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
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
of a synthesized inductor can be selected with ease. The Q
Q factor
In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
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.
Grounding. The fact that one side of the simulated inductor is grounded restricts the possible applications (real inductors are floating). This limitation precludes its use in low-pass and notch filters, leaving high-pass and band-pass filters as the only possible applications.
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, RLCRLC circuit
An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively...
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
Plain old telephone service
Plain old telephone service is the voice-grade telephone service that remains the basic form of residential and small business service connection to the telephone network in many parts of the world....
system. This has allowed telephones to be much smaller, as the gyrator circuit carries the DC
Direct current
Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...
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. Circuitry in telephone exchanges has also been affected with gyrators being used in line card
Line card
A line card or Digital Line Card is a modular electronic circuit on a printed circuit board, the electronic circuits on the card interfacing the telecommunication lines coming from the subscribers to the rest of the telecommunications access network.A line card commonly interfaces the twisted pair...
s. 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
Pilot signal
In telecommunications, a pilot is a signal, usually a single frequency, transmitted over a communications system for supervisory, control, equalization, continuity, synchronization, or reference purposes....
filters.
There are many applications where it is not possible to use a gyrator to replace an inductor:
- High voltageHigh voltageThe term high voltage characterizes electrical circuits in which the voltage used is the cause of particular safety concerns and insulation requirements...
systems utilizing flyback (beyond working voltage of transistors/amplifiers) - RF systems (RF inductors are usually small anyhow)
- Power conversion, where a coil is used as energy storage.
See also
- Negative impedance converterNegative impedance converterThe negative impedance converter 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...
(which can be used to implement a negative inductor with a capacitor) - Sallen–Key topology
External links
- Good description of this form of the simulated inductor — Elliot Sound Products
- Another description, with the same circuit
- LC filter design using equal value R gyrator, an alternative design
- An alternative circuit
- Webarchive backup: Another alternative circuit
- Discussion of the gyrator in general and a macro for Micro-Cap V
- Java simulation of this circuit
- Single transistor gyrator for telephony applications
- SPICE Analysis of gyrator for telephony applications
- Negative floating inductor with only 2 Op-amps Article here
