Side-by-side comparison
| Parameter | First Order | Second Order Active Filter |
|---|---|---|
| Roll-off rate | –20 dB/decade (–6 dB/octave) above f_c | –40 dB/decade (–12 dB/octave) above f_c |
| Transfer function order | First-order: H(s) = K·ω_0 / (s + ω_0) | Second-order: H(s) = K·ω_0² / (s² + (ω_0/Q)s + ω_0²) |
| Quality factor Q | Not applicable — only one pole, no resonance | Q determines passband shape: Q=0.707 Butterworth, Q>0.707 peaks |
| Op-amp count | One op-amp (buffer or low gain, e.g., non-inverting with R_f/R_1 setting gain) | One op-amp (Sallen-Key) or two op-amps (MFB topology) |
| Component count | One R, one C plus op-amp — minimal | Two Rs, two Cs plus op-amp (Sallen-Key); more for MFB |
| Gain-bandwidth interaction | Gain set independently; pole not shifted by gain change | In Sallen-Key, closed-loop gain affects Q — gain change shifts frequency response |
| Butterworth design | Q not defined — magnitude is 3 dB down at f_c by design | Butterworth 2nd order: Q = 0.707 (1/√2), gives maximally flat response |
| Topology examples | RC + voltage follower (unity gain), inverting integrator | Sallen-Key LPF (TI SLOA049 design), Multiple Feedback (MFB) BPF |
| Attenuation at 2×f_c | –7 dB (single pole) | –12 dB (double pole) — 5 dB more rejection |
| Real design IC | LM741, TL071 for audio low-pass at f_c = 1 kHz | UA741, TL082 Sallen-Key for anti-aliasing before a 10-bit ADC |
Key differences
At one decade above the cutoff, a first-order filter gives –20 dB (10× reduction in voltage); a second-order gives –40 dB (100× reduction). For an anti-aliasing filter before a 10-bit ADC sampling at 10 kHz, second order gets you from 80 dB of dynamic range to usable noise floor. The Q factor is unique to second order: Butterworth Q = 0.707 gives no passband ripple; Chebyshev Q > 0.707 adds ripple but steepens the transition. In the Sallen-Key topology, the passband gain K directly affects Q — raising gain by 3 dB doubles Q and can make the filter oscillate if K reaches 3 (for second-order Sallen-Key LPF with equal R and C).
When to use First Order
Use a first-order active filter when a gentle roll-off is sufficient and component count matters — for example, a 1 kHz low-pass built with one 16 kΩ resistor, one 10 nF capacitor, and a TL071 buffer removes audio-band noise with minimal board space.
When to use Second Order Active Filter
Use a second-order active filter when steeper roll-off or a specific Q is required — for example, a Sallen-Key Butterworth LPF at 5 kHz (Q = 0.707) as an anti-aliasing filter before a 12-bit ADC sampling at 44.1 kHz in an audio digitiser.
Recommendation
For exam filter design problems, identify the required roll-off first. If the question specifies –40 dB/decade or mentions Butterworth or Chebyshev, choose second-order. If simplicity and a single pole are enough, choose first-order and design with a single RC pair plus a non-inverting op-amp buffer.
Exam tip: Examiners ask you to derive the transfer function of a Sallen-Key LPF and identify the conditions for Q = 0.707 (Butterworth) — know that for equal R and C, the gain must be set to K = 1.586 (i.e., R_f/R_1 = 0.586) to achieve Butterworth response.
Interview tip: Interviewers at analog IC companies or DSP teams ask you to explain why a higher Q can make a Sallen-Key filter oscillate — answer that when K = 3 in the standard equal-R equal-C Sallen-Key LPF, the denominator's s-term coefficient goes to zero, giving a pole pair exactly on the j-omega axis.