The Mathematics
For a transfer function H(s), the Bode plot evaluates H(jω) for ω ∈ (0, ∞):
- Magnitude:
20·log₁₀|H(jω)|in decibels (dB). - Phase:
arg(H(jω))in degrees.
Second-order low-pass
H(s) = ωc² / (s² + 2ζωc·s + ωc²)
The damping ratio ζ controls behaviour near ωc:
- ζ = 1: Critically damped, smooth −40 dB/decade rolloff.
- ζ < 1/√2: Underdamped — a resonant peak appears before rolloff.
- ζ → 0: Pure resonance — a sharp spike at ωc.
Filter types
- Low-pass (LP): Pass low frequencies, reject high. Anti-aliasing, smoothing.
- High-pass (HP): Pass high, reject low. AC coupling, edge detection.
- Band-pass (BP): Pass a frequency band, reject everything else. Radio tuning.
- Notch: Reject a narrow band. Power-line interference removal (50/60 Hz).
Further reading
- Bode plot (Wikipedia)
- Oppenheim & Willsky, Signals and Systems
- Ogata, Modern Control Engineering