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A gyrator or positive impedance inverter is an electric circuit which 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 bandpass filter behave like a band-stop filter, and so on. It is primarily used in active filter design and miniaturization.

Generic gyrator

An ideal gyrator is a two port device that provides the following transmission matrix relationship between the instantaneous voltages and currents at its ports:

${\displaystyle {\begin{aligned}{\begin{bmatrix}v_{1}\\i_{1}\end{bmatrix}}=&{\begin{bmatrix}v_{2}\\-i_{2}\end{bmatrix}}\\=&{\begin{bmatrix}0&1/g_{2}\\g_{1}&0\end{bmatrix}}{\begin{bmatrix}v_{2}\\-i_{2}\end{bmatrix}}\end{aligned}}}$

where g₁ and g₂ are known as the gyration conductance, measured in siemens. Consequently, the current-voltage characteristic presented on one port is inversely proportional to that of the load present on the other port:

${\displaystyle \ {\frac {v_{1}}{i_{1}}}=-{\frac {1}{g_{1}g_{2}}}{\frac {i_{2}}{v_{2}}}}$

If g₁ ≠ g₂, then the gyrator can only be implemented with an active circuit. If, however, they are equal then the gyrator can be implemented with an entirely passive network.

If one of the ports is connected to a linear load, the gyrator's other port presents an impedance inversely proportional to that of the physical load,

${\displaystyle \ Z_{\mathrm {in} }={\frac {1}{g_{1}g_{2}}}{\frac {1}{Z}}}$

Implementation: a simulated inductor

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, 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.

There are a variety of transistor and op-amp gyrator implementations. A simple simulated inductor based on a unity-gain amplifier is presented in the figure on the left below.

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 idea of the circuit is revealed in the figure on the right by an equivalent balanced bridge circuit. It consists of two legs connected in parallel: the left leg is the "shaping" RC circuit; the right leg is the virtual inductor V_L with its internal resistance R_L.

In balanced bridge circuits the voltages across the opposite elements are equal. Here, the varying voltage source Vs (the op-amp follower) balances the bridge by "copying" the voltage drop across the resistor R that is proportional to the current flowing through the capacitor. As a result, the voltage drop across the resistor R_L is equal to the voltage across the capacitor; so, the current flowing through R_L is proportional to the voltage across the capacitor. The desired effect is an impedance of the form of an ideal inductor L with a series resistance R_L:

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

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 R_LRC = 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 adequately large that the dominant term is; so, the RC circuit does not impact the input:

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

This is the same as a resistance R_L in series with an inductance L = R_LRC. There is a practical limit on the minimum value that R_L can take, determined by the current output capability of the op amp.

Although this is a positive feedback circuit, it is always stable because of the unity-gain amplifier.

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 synthetic 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 (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 flying). 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, 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. 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 (RF inductors are usually small anyhow)
• Power conversion, where a coil is used as energy storage.

See also

• Negative impedance converter (which can be used to implement a negative inductor with a capacitor)
• Sallen–Key topology

References

1. http://inst.eecs.berkeley.edu/~ee100/fa04/lab/lab10/EE100_Gyrator_Guide.pdf
2. Theodore Deliyannis, Yichuang Sun, J. Kel Fidler, Continuous-time active filter design, pp.81-82, CRC Press, 1999 ISBN 0849325730.
3. An audio circuit collection, Part 3

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

• Analog circuits