The Gyrator: Faking a Henry-Sized Inductor With One Op-Amp and a Capacitor

What a Gyrator Is and Where You Find One

The gyrator was proposed by Dutch engineer Bernard D. H. Tellegen in 1948 as the fifth ideal linear circuit element, joining the resistor, capacitor, inductor, and transformer. Unlike those four, it is non-reciprocal (antireciprocal): it defines a relationship where the voltage on one port is set by the current on the other, so it can invert an impedance rather than just scale it.

Its most useful job is inductance simulation. Terminate one port of a gyrator with a capacitor and the other port looks like an inductor. This matters because at audio and sub-megahertz frequencies, real inductors are physically enormous, lossy, pick up hum, and cannot be integrated on silicon.

• Graphic and parametric equalizers — each frequency band is a resonant LC section; gyrators replace the coils.
• Active LC-ladder filters — telephone line equalizers and RF-block filters.
• Oscillators and phantom-power circuits — where a large simulated inductance sets frequency or blocks AC.

The gyrator also underlies the isolator and circulator concepts in microwave engineering, though there magnetic ferrites, not op-amps, do the gyration.

How the Simulation Works: Impedance Inversion

The core identity of an ideal gyrator with gyration resistance R_g is Z_in = R_g² / Z_load. Load it with a capacitor, Z_load = 1/(jωC), and the input impedance becomes Z_in = R_g² · jωC = jω(R_g²C). That is precisely the impedance of an inductor jωL with L = R_g²C — a capacitance has been turned inside-out into an inductance.

The popular single-op-amp version does the same thing with a subtle asymmetry. An input resistor R1 feeds the node; the op-amp acts as a unity-gain buffer sensing the voltage across an R2–C divider to ground. At low frequency the capacitor is open, the buffer holds R1's far end near the input, and little current flows — high impedance, like an inductor. As frequency rises, C shorts, the buffer pulls the node, and current climbs. The node current lags the applied voltage, mimicking an inductor. Working through the node equations gives:

• L_eq = R1 · R2 · C — the simulated inductance
• R_s = R1 — an unavoidable series loss resistance in series with L_eq

So the port behaves as an ideal inductor L_eq in series with resistor R1, i.e. Z ≈ R1 + jω(R1·R2·C).

Key Quantities and a Worked Example

Suppose you need a 10 H simulated inductor for a 1 kHz equalizer band. Pick a common film capacitor C = 100 nF and R2 = 100 kΩ. Then:

• L_eq = R1 · R2 · C ⇒ R1 = L / (R2·C) = 10 / (10⁵ × 10⁻⁷) = 1 kΩ.

At 1 kHz the inductive reactance is X_L = 2πfL = 2π·1000·10 ≈ 62.8 kΩ. The quality factor is Q = X_L / R_s = ωL / R1 = 62.8k / 1k ≈ 63 — comparable to a good physical coil, and tunable simply by lowering R1.

The self-resonant frequency is set by whatever capacitance the gyrator resonates against; unlike a real coil, the simulated inductor has almost no parasitic winding capacitance, so the resonance is clean and predictable. Note the design trade: raising R2 to hit a large L keeps R1 small (good for Q) but pushes the op-amp toward its input-bias-current and gain-bandwidth limits. Typical realizable values span 1 mH to over 1000 H, far beyond what any practical wound part offers at these frequencies.

Designing and Operating One in Practice

Good gyrator design is mostly about respecting the op-amp's limits and choosing components that keep the loss term honest.

• Pick C first. Use a stable film or C0G ceramic (not high-K X7R, whose capacitance drifts with voltage and temperature), because L_eq tracks C directly.
• Set R2 for the inductance, R1 for the Q. Because R_s = R1, a smaller R1 gives higher Q — but R1 also sets the DC path resistance, so it cannot go arbitrarily low without loading the source.
• Mind the gain-bandwidth product (GBW). An op-amp's finite GBW and slew rate cap useful operation near 100 kHz; above that the simulated L develops phase errors and can even look negative-resistive, causing oscillation.
• Keep bias current in check. Large R2 (hundreds of kΩ) means a FET-input op-amp (low I_bias) is preferred to avoid DC offset.

Ground one terminal — that is mandatory (see limitations). Bypass the supplies well, since any noise on the rails couples through the buffer into the simulated coil and shows up as excess series loss or hum in an audio band.

Gyrator vs. GIC vs. a Real Coil

The single-op-amp gyrator is the cheapest simulated inductor but is inherently lossy (R_s = R1). For applications demanding a near-lossless, high-Q inductor, engineers reach for the Antoniou Generalized Impedance Converter (GIC), published by A. Antoniou in 1969.

• Antoniou GIC uses two op-amps and five impedances. Its input impedance is Z_in = (Z1·Z3·Z5)/(Z2·Z4); making Z4 a capacitor and the rest resistors yields L = (R1·R3·R5·C4)/R2 with, ideally, infinite Q — no fixed series-loss term. It is the workhorse of high-order active-LC ladder filters.
• Real wound inductor wins only above ~1 MHz and whenever a truly floating inductor is required, since it stores real magnetic energy and needs no power supply.

Compared with an RC active filter (Sallen-Key, MFB), the gyrator approach directly mimics a passive LC prototype, inheriting its low component-value sensitivity — a big reliability advantage for steep, high-order filters where op-amp-only topologies become touchy.

Failure Modes, Trade-offs, and Significance

The gyrator's defining limitation is the grounded terminal: because the op-amp derives its reference from ground, the simulated inductor has one end tied to ground and cannot float. That rules it out of low-pass and series-notch topologies that need a floating series coil — one of the few places a real inductor is still irreplaceable without extra circuitry.

• Bandwidth ceiling. Finite GBW and slew rate confine it to roughly DC–100 kHz; push it higher and the phase error can turn the port into a negative resistance, spawning parasitic oscillation.
• Powered, not passive. It needs a supply; on power loss the "inductor" simply vanishes, unlike a coil.
• Noise and headroom. Op-amp voltage noise and clipping set the dynamic range; a real coil has none of that.

Despite these caveats, the gyrator is one of analog design's great enabling tricks. It made bulky, hum-prone, un-integrable inductors disappear from equalizers, filters, and telephone line cards, and it embodies a deep idea — that with an active element you can synthesize any two-terminal immittance you like, coil or not.

Real wound inductor vs. single-op-amp gyrator vs. two-op-amp Antoniou GIC

Property	Wound inductor	Single-op-amp gyrator	Antoniou GIC (2 op-amps)
Energy storage	Magnetic field in coil	Electric field in capacitor	Electric field in capacitor
Practical value range	nH – tens of mH	1 mH – 1000+ H	1 mH – 1000+ H
Series loss / Q	Q set by wire, 30–300	R_s = R1 fixed by design	Ideal Q → ∞ (lossless)
Floating (both ends free)	Yes	No — one end grounded	No — one end grounded
Frequency ceiling	GHz (air-core)	~100 kHz (GBW/slew limited)	~100 kHz–1 MHz
Cost / size at 10 H	Huge, heavy, ~$$$	Op-amp + 3 parts, tiny	2 op-amps + 5 parts, tiny