Techniques for higher order active filters

Higher-order active filters (order > 2) are designed by one of these methods:

1. Cascade design

The transfer function is factored into second-order sections (plus one first-order section for odd order), each realised by a biquad (Sallen–Key, MFB, Tow–Thomas, KHN) and connected in cascade. Op-amp outputs have low impedance, so sections do not load each other.

• Simple to design and tune; each pole pair is set separately.
• Sensitivity is higher, especially for high-Q sections.
• Order of sections: usually low Q first, high Q last.

2. Direct (element) simulation of the LC ladder

Start from the doubly terminated LC ladder, which has the lowest sensitivity, and replace the inductors:

• Gyrator / GIC inductor simulation: grounded inductors (highpass ladders) by GICs, floating ones by two GICs or gyrators.
• FDNR (Bruton transformation): divide impedances by $s$; inductors become resistors and capacitors become grounded FDNRs (lowpass ladders).

3. Operational simulation (leapfrog)

The state equations of the ladder (currents and voltages of L and C) are realised with integrators and summers that feed each other alternately. It keeps the ladder's low sensitivity and suits lowpass and bandpass filters.

4. Multiple-loop feedback (MLF)

Integrator or biquad blocks with several feedback paths, e.g. follow-the-leader feedback (FLF) and primary-resonator block (PRB); the leapfrog is also a multiple-loop structure.