# Gyrator

The gyrator is an electric circuit which inverts an impedance. In other words, it can make a capacitive circuit behave inductively, a bandpass filter behave like a band-stop filter, and so on. It was invented around 1948 by B.D.H. Tellegen of Philips Research Laboratories, Eindhoven ("The gyrator, a new electric network element", Philips Res. Rep. 3 (1948) pgs 81-101) and is used in active filter design.

## Simulated Inductor

Gyrator simulating Inductance

Its primary use is to simulate an inductive element in a small electronic circuit or integrated circuit, which is typically much smaller than a tabletop circuit. Before the invention of the transistor, coils of wire, with large inductance, might be used in electronic filters, for example. Thus a real inductor can be replaced by a much smaller assembly containing a capacitor, operational amplifiers or transistors, and resistors, which is especially useful in integrated circuit technology. In addition, real capacitors are often much closer to "ideal capacitors" than real inductors are to "ideal inductors". Because of this, a synthetic inductor realized with a gyrator and a capacitor may, for certain applications, also be closer to an "ideal inductor" than any real inductor can be. Thus, use of capacitors and gyrators may improve the quality of filter networks that would otherwise be built using inductors. In addition, as shown below, the Q factor of the synthesized inductor can be selected with ease.

## Operation of the circuit

The circuit works by inverting the effect of the capacitor. The desired effect is an impedance of the form

[itex]Z = R_L + j \omega L \,\![itex]

This is an ideal inductor L with a series resistance RL. From the diagram, it can be seen that the impedance of the simulated inductor is the desired impedance in parallel with the impedance of C and R.

[itex]Z_{in} = \left( R_L + j \omega R_L R C \right) \| \left( R + {1 \over {j \omega C}} \right) [itex]

If R is much greater than RL, this comes close to

[itex]Z_{in} = R_L + j \omega R_L R C \,\![itex]

This is the same as a resistance RL in series with an inductance L = RLRC. It differs in function from a true inductor due to the parallel RC term, and because RL is large compared to a real inductor. A real inductor has low internal resistance caused only by the wire it is made of. This limits the Q factor, or accuracy, of filters that can be made with the simulated inductor.

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