Fourth-order Low-Pass Filter with Floating Gyrator

Charles Eric LaForest, PhD., GateForge Consulting, Ltd.

This is a sketch of a fourth-order low-pass filter circuit. However, instead of the usual multiple-feedback or Sallen-Key designs we use a plain RLC design with the lesser-known floating gyrators to synthesize the inductors. An RLC filter's parameters are very easy to calculate and its behaviour very easy to understand.

There is also an equivalent High-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.

The implementation for the floating gyrator comes from a hint on the Gyrator Wikipedia page:

"However the gyrator can be used in a floating configuration with another gyrator so long as the floating "grounds" are tied together. This allows for a floating gyrator, but the inductance simulated across the input terminals of the gyrator pair must be cut in half for each gyrator to ensure that the desired inductance is met (the impedance of inductors in series adds together)."

You can try out this circuit (including the bias supply and a pre-biased signal source) in the interactive CircuitJS1 simulator: floating_gyrator_lpf.cjs1.

Design and Calculations

The active and passive circuits on the left both implement a fourth-order low-pass filter (LPF), as a buffered pair of second-order LPFs, 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 inductance (L) is 100mH, which is closely approximated by (R1a+R1b)*C1*R2.
• R1a and R1b (100Ω), summed together as R1 (200Ω), are the intrinsic resistance of the inductor
• R2 (100kΩ) represents the eddy current losses in the inductor
• C1 (10nF) is the inductor's parasitic capacitance
• C2 (100nF) is the filter capacitor
• R3 (247Ω) (along with R1) controls the Q of the whole filter (not the Q of the inductor itself, which is set by R1, R2, and C1, and isn't relevant here)
• The filter critical frequency (Fc) is 1/(2*π*sqrt(L*C2)), 1592 Hz here.
• The filter Qs (series Q) is approximated by (1/(R1+R3))*sqrt(L/C2), 2.24 here. The actual Qs will be slightly lower due to the losses from C1 and R2, and any unaccounted impedance in the signal source which is thus in series with R3.

Filter Design Constraints

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:

• The inductance should be as large as necessary to keep C2 small, and thus the maximum Qs large, which can then be moderated by R3. A larger R3 means less current at the op amp output.
• The Qs of a HPF is also the peak resonance gain of the filter at Fc. Since we have two filters in series, both filter's Qs will multiply together to the final overall Qs. Making them both equal to sqrt(Qs_overall) is convenient and maximizes the range of input levels to each filter before clipping.

Gyrator Design Constraints

By comparing the gyrator's synthesized inductance ((R1a+R1b)*C1*R2) with the equivalent passive circuit, some constraints emerge:

• C1 cannot be so small as to be affected by parasitic capacitances.
• R1a and R1b must be equal and should be as small as possible to maximize Qs and make R2 and R3 as large as possible, but how small is limited by how much current the op amp output will need to sink or source, depending on R3 and the source impedance (which should be low to avoid limiting Qs).
• R2 should be as large as possible to limit simulated eddy current losses and thus improve the accuracy of the Qs approximation, but not so large as to become sensitive to noise (Johnson or external).
• A larger R3 means more range to control the Qs, since we cannot alter R1 without altering the inductance.

Gyrator Impact on Bias Current

Unfortunately, unlike in the single-supply gyrator-based High-Pass Filter, the capacitor C2 is the path to the virtual ground (Vbias), and so there are no active devices to take away the current load from the bias voltage source. It is thus critical for the bias voltage source to be buffered.