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 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. 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. 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
where is the gyration resistance of the gyrator.
The gyration resistance 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, 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
direction, but reverses the polarity of the signal travelling in the backward
direction. The symbol used to represent a gyrator in one-line diagrams, 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,
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. Name
Tellegen named the element gyrator as a portmanteau of gyroscope and the common
device suffix -tor 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.
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. 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. Application: a 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 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 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; 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 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. This
limitation may preclude its use in some low-pass and notch filters. 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. 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. Gyrators are used in
most DAAs. 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
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. A passive
gyrator that does not have the drawback of negative elements is the quarter-wave
impedance transformer which is widely used in microwave circuits. This is an
example of a distributed element circuit.
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. 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. 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.
See also Sallen–Key topology
References Berndt, D. F.; Dutta Roy, S. C.,
"Inductor simulation with a single unity gain amplifier", IEEE Journal of Solid
State Circuits SC–4: 161–162 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