Charles Eric LaForest, PhD., GateForge Consulting, Ltd.
This is a sketch of a fourth-order high-pass filter circuit. However, instead of the usual multiple-feedback or Sallen-Key designs we use a plain RLC design with conventional grounded gyrators to synthesize the inductors. An RLC filter's parameters are very easy to calculate and its behaviour very easy to understand. It also demonstrates a non-obvious design feature of gyrators which greatly reduces biasing voltage distortion.
There is also an equivalent Low-Pass Filter.
This material is almost entirely derived from Active Filters Using Gyrators - Characteristics, and Examples by Rod Elliott of Elliott Sound Products (thank you!). Go read it if you want to know how a gyrator works.
You can try out this circuit (including the bias supply and a pre-biased signal source) in the interactive CircuitJS1 simulator: gyrator_hpf.cjs1.
The active and passive circuits on the left both implement a fourth-order high-pass filter (HPF), as a buffered pair of second-order HPFs, with an overall calculated critical frequency (Fc) of 1592 Hz and a Q of 5. Both circuits are exactly equivalent as long as the op amps operate in their linear region (no clipping, current limiting, slew rate limiting, etc...) and input current is negligible (e.g.: TL072).
For each second-order filter in both circuits:
The filter's design brings in some constraints due to the limited input/output range of the op amps, and the nature of RLC circuits at audio frequencies:
By comparing the gyrator's synthesized inductance (R1*C1*R2) with the equivalent passive circuit, some constraints emerge:
One useful and non-obvious difference between the passive and active designs is the path the signal current takes to the ground reference. This become significant in single-supply circuits where the virtual ground is provided by a bias voltage source (Vbias), as shown here, since Vbias connects to many points in a circuit and one excessive load will cause distortion at all points.
In the passive design, assuming high impedance inputs, the signal current has to be entirely sourced or sunk by the Vbias source, which can be several milliAmps. Even with a buffered source, this is a significant load.
In the active design, the gyrator op amp output sources or sinks the signal current, meaning that current ends up at the well-decoupled supply pins, is distributed over multiple op amps, and only a few microAmps are needed from the Vbias source since R2 is so large.