How it works
A 2nd-order active BPF is formed by cascading a low-pass and high-pass response; the passband is the frequency range where both sections allow signal through simultaneously. For the Multiple Feedback topology: f_0 = (1/2π)·√((R1+R3)/(R1·R2·R3·C²)); Q = (f_0·2π·C·√(R1·R2·R3/(R1+R3))); midband gain A_0 = −R2/(2R1). With equal capacitors C1 = C2 = C, the design procedure starts by selecting C, then calculating resistors from Q, f_0, and A_0 specifications. Op-amp slew rate must satisfy SR ≥ 2π·f_0·V_out_peak to avoid distortion at centre frequency.
Key points to remember
Q factor = f_0 / BW, where BW = f_H − f_L is the 3 dB bandwidth. Higher Q means sharper selectivity and slower transient response. For the MFBF topology, maximum practical Q is about 20–25 before component sensitivity and op-amp gain-bandwidth product become limiting factors. The Wien bridge notch filter is the complement of the Wien bridge band-pass. Centre frequency can be tuned by simultaneously scaling both capacitors; Q can be adjusted by changing R3 without affecting f_0. A state-variable filter (using three op-amps) provides simultaneous LP, BP, and HP outputs and allows independent adjustment of f_0, Q, and gain.
Exam tip
The examiner always asks you to derive the expressions for centre frequency f_0 and Q for the multiple feedback band-pass filter topology — set up the nodal equations in the s-domain and identify the standard 2nd-order BPF form H(s) = K·s·ω_0/Q / (s² + s·ω_0/Q + ω_0²).