Inductors are the problem child of analog design: they're bulky, expensive at high values, have parasitic resistance and capacitance, pick up magnetic interference, and don't integrate onto silicon. A gyrator sidesteps all of this by using an op-amp, a capacitor, and a couple of resistors to synthesize inductive behavior. The circuit doesn't store energy in a magnetic field — it stores it in the capacitor — but at its input terminals it looks electrically like a grounded inductor.

The classic single-op-amp gyrator (Antoniou's simulated inductor) has this topology: input signal feeds R1 to the op-amp's inverting input and also to one terminal of capacitor C. The op-amp's output drives the other terminal of C through R2, and its non-inverting input ties to the input node. The op-amp forces its output to whatever voltage maintains zero differential input, and the result is an input impedance:

Z_in = R1 + jω(R1·R2·C)

The imaginary part behaves exactly like an inductor of value L_eq = R1·R2·C, with R1 appearing as series DC resistance.

Worked example: Want a 10 H "inductor" for a subwoofer-frequency notch filter? Pick C = 1 µF, R1 = 1 kΩ, R2 = 10 MΩ. That gives L_eq = 1k × 10M × 1µ = 10 H. A real 10 H inductor would be the size of a fist, weigh half a pound, and cost $40. The gyrator version fits in a TO-92 footprint and costs under a dollar.

Where you'll actually see this:

• Graphic equalizers — the sliders in old hi-fi EQs almost always used gyrators to synthesize the big inductors needed for bass-band peaking filters.
• Telephone line filters — replacing audio-frequency chokes in DAA circuits.
• Subsonic and seismic filters where the required L would otherwise be impractical.

The catches you need to respect:

• It only simulates a grounded inductor. One terminal must go to ground — you can't insert it freely in series like a real coil. (Floating gyrators exist but need two op-amps.)
• The op-amp's gain-bandwidth product limits the upper frequency. Above roughly GBW/100, the simulated L degrades and can even go inductive-with-loss or worse. Stick to audio and sub-audio bands with garden-variety op-amps.
• It can't handle large currents or store significant energy — don't try to build a power-supply choke this way.
• Q is limited by R1 (the unavoidable series resistance). Lower R1 means higher Q but more loading on the source.

Rule of thumb: For audio-band synthetic inductors up to ~10 H, pick C in the 10 nF–1 µF range, then choose R1·R2 to land on your target L. Keep R2 ≤ 10 MΩ to avoid bias-current errors, and use a FET-input op-amp (TL072, OPA2134) for clean operation.