Op-Amp Filter Design Fundamentals Analog Signal Processing

2nd-Order Sallen-Key Active Low-Pass Filter

Tweak component values to shape Butterworth, Chebyshev, or Bessel frequency responses.

Demonstrates active op-amp filtering with -40 dB/decade roll-off. Watch how damping factor controls frequency peaking at the cutoff boundary and phase transition from 0° to -180°.

Governing Physical Law & Equations

H(s) = \frac{K \omega_c^2}{s^2 + \frac{\omega_c}{Q} s + \omega_c^2}, \quad f_c = \frac{1}{2\pi \sqrt{R_1 R_2 C_1 C_2}}

Second-order biquadratic transfer function with cutoff frequency and Q-factor.

Law: Negative Feedback & Complex Pole Placement | Standard Reference: IEEE Std 1057 / Butterworth-Chebyshev Biquad Filter Realization

Adjustable System Parameters

Parameter	Nominal Value	Dynamic Range	Physical Role
Corner Frequency (f_c)	1500 Hz	100 to 10000 Hz	-3dB transition frequency
Quality Factor (Q)	0.707	0.3 to 3	0.707 for Butterworth (maximally flat), >0.707 for Chebyshev ripple
Passband Gain (A_v)	1 V/V	1 to 5 V/V	Non-inverting op-amp closed loop gain
Input Test Signal (f_in)	1200 Hz	50 to 15000 Hz	Frequency of probe signal through the filter

Analytical Proof & Derivation

Nodal KCL analysis around the op-amp feedback loop yields characteristic polynomial s² + (ω_c / Q) s + ω_c². Corner frequency f_c = 1 / (2π √(R₁ R₂ C₁ C₂)). Setting Q = 1/√2 ≈ 0.7071 yields the maximally flat Butterworth response with zero passband ripple and asymptotic -40 dB/decade rolloff.

Verification Benchmark

Active filter benchmark: For f_c = 1500 Hz, Q = 0.707, theoretical attenuation at f = f_c is exactly -3.0103 dB with phase shift -90.0°. Computed filter response precisely reproduces the theoretical pole trajectory.

Field Engineering Insights

Resistor and capacitor component tolerances directly degrade Q. A 5% capacitor drift in a Q = 2 Chebyshev configuration can cause a 2.5 dB unexpected gain peak in the passband. In industrial designs, 1% metal film resistors and COG/NPO dielectric capacitors are mandated.