What Are Gyrators?

Gyrators are two-port electrical network elements that relate the voltage at one port to the current at the other port, effectively inverting the impedance seen at each terminal with respect to a characteristic conductance parameter called the gyration conductance. Defined theoretically by Bernard Tellegen in 1948 as the fifth fundamental linear two-port element alongside the resistor, capacitor, inductor, and transformer, the gyrator is passive, lossless, and non-reciprocal: a signal transmitted from port 1 to port 2 undergoes a 180-degree phase shift relative to the transmission in the reverse direction.

The gyrator's most consequential property is impedance inversion: a capacitive load connected to one port appears inductive at the other, and vice versa. This property makes the gyrator a tool for simulating inductors using capacitors and active circuitry, a capability of particular value in integrated circuit design where physical inductors are large, lossy, and difficult to fabricate.

Circuit Theory and Impedance Inversion

The gyrator is described by the two-port constitutive relation in which the port currents are proportional to the cross-port voltages via the gyration conductance G. For an ideal gyrator loaded at port 2 with a capacitor C, the impedance seen at port 1 is that of an inductor with value L = C/G^2. This transformation is exact and lossless for an ideal gyrator, making it mathematically equivalent to a physical inductor with no series resistance and no core loss.

Research on practical gyrator realizations published in IEEE Transactions on Circuit Theory established key design rules for gyrator circuits built from operational amplifiers and resistors, showing that two cross-coupled transconductance stages can approximate the ideal element to the bandwidth limit of the amplifiers. Work on gyrators in two-port network theory situates the gyrator alongside the ideal transformer as one of two canonical power-transferring two-ports, clarifying the energy-flow relationships that distinguish it from purely dissipative elements.

Ferrite-based physical gyrators exist at microwave frequencies, where the non-reciprocal permeability of a magnetized ferrite material naturally produces the asymmetric transmission required by the gyrator's definition. These ferrite gyrators are used in circulators and isolators for signal routing in waveguide and coaxial systems, though they are distinct in construction from the active electronic gyrators used in audio and low-frequency circuits.

Active Inductor Realizations

The principal engineering application of the gyrator concept is the synthesis of inductors on integrated circuits. A capacitor terminated gyrator, built from two transconductance amplifiers with cross-coupled feedback, presents an inductive impedance at its input terminals over the bandwidth where the amplifiers maintain adequate gain and phase margin. These active inductors avoid the need for on-chip spiral inductors, which suffer from low quality factors at frequencies below several gigahertz due to substrate eddy current losses.

IEEE conference research on gyrator-based frequency-tuning circuits demonstrates active inductors built in CMOS technology that achieve tunable inductance values and quality factors suitable for bandpass filter applications in the low-gigahertz range. The tuning is achieved by adjusting the bias currents of the transconductance stages, varying the effective gyration conductance and therefore the simulated inductance.

In power electronics, gyrator-based circuits have been studied for impedance matching and power conditioning, with applications in resonant converters where precise reactive element values are required but physical inductors add cost, weight, and core-saturation concerns.

Applications

Gyrators have applications across several areas of circuit design and microwave engineering, including:

• On-chip inductor synthesis in RF integrated circuits and oscillators
• Ladder filter design using capacitors to simulate LC networks
• Microwave circulators and isolators based on ferrite gyrator properties
• Active resonator circuits for bandpass filtering in wireless transceivers
• Nonlinear circuit synthesis where controllable gyrators replace passive elements