Second-order low-pass filter. Flat at low frequencies, −40 dB/decade rolloff after cutoff. Phase goes 0° → −180°.
The Core Idea
A Bode plot is the frequency-domain fingerprint of a system. It shows how much a sinusoidal input is amplified or attenuated at each frequency (magnitude), and how much the output is shifted in phase (phase). Together, they tell you everything about how a filter, amplifier, or control loop behaves across the spectrum.
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.
Second-order high-pass. Signal rejection below ωc, +40 dB/decade rise, flat above. Phase: +180° → 0°.
Switch between filter types to see how the same second-order structure produces opposite behaviours depending on topology:
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).
Band-pass filter. Only frequencies near ωc pass through — the hallmark of a resonant system.
Stability from Bode plots
For feedback with open-loop G(s):
- Gain margin: How much gain increase before instability (measured at phase = −180°).
- Phase margin: How much phase lag before instability (measured at gain = 0 dB).
Rule of thumb: phase margin > 45° for comfortable stability.
Further reading
- Bode plot (Wikipedia)
- Oppenheim & Willsky, Signals and Systems
- Ogata, Modern Control Engineering