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Frequency Response: Bode and Nyquist Plots For students.

EngineeringTry the interactive 3D model →

Frequency response is a system's steady-state reaction to sinusoids of different frequencies. A Bode plot shows magnitude (dB) and phase (degrees) versus log frequency as easy hand-sketched asymptotes. A Nyquist plot shows the same data as one polar curve of the open-loop transfer function as frequency sweeps 0 to infinity.

Stability margins read directly off open-loop data, even measured experimentally from hardware with unknown equations. Gain margin and phase margin are the industry robustness numbers: how much extra gain or lag the loop tolerates before oscillating. Amplifier, power-supply and process engineers live on Bode plots.

Key Concepts

  • Corner frequency: where a first-order term bends the asymptote, e.g. 1/tau for 1/(tau x s + 1).
  • Slope rules: each pole: -20 dB/decade, -90 degrees; each zero: +20 dB/decade, +90 degrees; 1/s starts at -20 dB/decade, -90 degrees.
  • Gain crossover: magnitude crosses 0 dB. Phase crossover: phase crosses -180 degrees.
  • Gain margin: tolerable gain rise, measured at phase crossover: GM = 1/|G(j x omega_pc)|.
  • Phase margin: tolerable extra lag, at gain crossover: PM = 180 + arg(G(j x omega_gc)).
  • Nyquist criterion: Z = P - N (Z = unstable closed-loop poles, P = unstable open-loop poles, N = net counterclockwise encirclements of -1).

Formulas

  • dB magnitude: 20 log_10(|G(j x omega)|).
  • GM: 1/|G(j x omega_pc)| (or -20 log_10 of it, in dB).
  • PM: 180 degrees + arg(G(j x omega_gc)).
  • Nyquist: Z = P - N; with P = 0, stability needs no net encirclement of -1.
  • Assumptions: LTI system; margins are cleanest for minimum-phase loops.

Worked Example

Phase margin of G(s) = 10/(s x (s+1) x (s+10)):

  1. Corners at 1 and 10 rad/s; 1/s gives initial -20 dB/decade.
  2. Gain crossover: at omega = 3, |G| = 10/(3 x sqrt(10) x sqrt(1.09)) = 10/9.905 = 1.0096 (~0.08 dB), so omega_gc ≈ 3 rad/s.
  3. Phase at 3: -90 - arctan(3) - arctan(0.3) = -90 - 71.57 - 16.70 = -178.27 degrees.
  4. PM = 180 - 178.27 = 1.73 degrees: barely stable. Any extra lag or gain oscillates near 3 rad/s. A designer would add lead compensation or cut gain for a comfortable 45-60 degrees.

Common Misconceptions

  • "You need the transfer function." Bode plots can be measured: inject sines, record gain/phase per frequency. Real hardware is analyzed this way.
  • "Positive margins always mean stable." True for ordinary minimum-phase loops, but exotic non-minimum-phase systems can still be unstable; Nyquist is the general arbiter.
  • "Nyquist equals root locus." Root locus tracks closed-loop poles vs gain; Nyquist maps open-loop frequency response. Different plots, different questions.

Real-World Applications

  • Op-amps: compensation for 45+ degrees phase margin.
  • Power supplies: loop gain shaped via Bode for load-step stability.
  • Audio: feedback-amplifier stability against oscillation.
  • Process control: margins verified after PID tuning for robustness.

Bode plots make frequency response hand-sketchable, with gain and phase margins reading straight off; Nyquist wraps it into one polar curve under Z = P - N. The example's 1.7-degree margin showed a loop on oscillation's edge. Design for margin, not mere stability, and systems survive the real world.

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