Bode Plot

Bode plots show gain (dB) and phase versus log frequency. Learn asymptotic sketching, corner frequencies, bandwidth, gain and phase margins from the open-loop plot, and loop-shaping design.

Feed a sine wave into a linear system and you get a sine wave out at the same frequency, scaled and phase-shifted. The Bode plot shows that scaling (magnitude, in decibels) and shift (phase) across frequency on logarithmic axes.

Because logarithms turn multiplication into addition, each pole and zero adds a simple straight-line contribution, so Bode plots can be sketched by hand. Plotted for the open loop, they directly show how stable the closed loop will be.

An analogy

A Bode plot is like a hi-fi equaliser display: it shows how much each frequency is boosted or cut, and how much it's delayed.

Reading the plot

Horizontal axis: frequency on a log scale (each decade ×10). Magnitude in dB: 0 dB = gain 1, +20 dB = ×10, −20 dB = ×0.1. Phase in degrees: negative means the output lags.

Asymptotic sketching

Each factor adds a straight-line contribution to the plot.

• Constant K: flat line at 20 log K, 0° phase.
• Integrator 1/s: −20 dB/dec through 0 dB at ω = 1, −90° phase.
• Pole 1/(τs + 1): flat, then −20 dB/dec above ω = 1/τ; phase goes 0 → −90° (−45° at corner).
• Zero: the mirror image (+20 dB/dec, +90°).
• Second-order poles: −40 dB/dec with a resonant peak when ζ < 0.7.

Stability margins

On the open-loop Bode plot, find where magnitude crosses 0 dB (ω_c): the phase there plus 180° is the phase margin. Find where phase crosses −180°: the magnitude there, negated, is the gain margin. Typical targets: PM 45–60°, GM > 6 dB.

Loop shaping

High loop gain at low frequency gives tracking and disturbance rejection; low gain at high frequency rejects noise and unmodelled dynamics; a slope of about −20 dB/dec near crossover gives good phase margin. Lead compensators add phase near ω_c; lag compensators raise low-frequency gain; PI adds an integrator below crossover.

Bandwidth and speed

Closed-loop bandwidth (−3 dB) is roughly the open-loop crossover frequency, and rise time ≈ 0.35/f_BW. Faster loops need higher crossover — limited by actuator saturation, sensor noise, delay and sampling rate (crossover typically < f_s/10).

Formulas

• Magnitude in dB: |G|_dB = 20·log₁₀|G(jω)|
• Pole at 1/τ: −20 dB/decade above corner, phase → −90° (−45° at corner)
• Phase margin: PM = 180° + ∠L(jω_c)
• Gain margin: GM = −|L(jω_180)|_dB
• PM ↔ damping (approx.): ζ ≈ PM / 100 (PM in degrees, PM < 70°)

Worked example: Phase margin of L = 100/(s(s + 10))

Given: L(jω) = 100/(jω(jω + 10))

Find: Gain crossover and phase margin.

• |L| = 100/(ω√(ω² + 100)) = 1 → ω⁴ + 100ω² − 10⁴ = 0
• ω² = (−100 + √(10⁴ + 4 × 10⁴))/2 = 61.8 → ω_c = 7.86 rad/s
• ∠L = −90° − arctan(7.86/10) = −128.2°
• PM = 180 − 128.2 = 51.8° (≈ ζ 0.5)

Answer: ω_c = 7.86 rad/s, PM ≈ 52°, GM = ∞

Worked example: Reading a single pole

Given: G(s) = 10/(0.1s + 1)

Find: Low-frequency gain, corner, and gain at 100 rad/s.

• DC: 20 log 10 = 20 dB
• Corner at 1/0.1 = 10 rad/s (17 dB, −45°)
• At 100 rad/s: one decade above → ≈ 20 − 20 = 0 dB, phase ≈ −84°

Answer: 20 dB flat to 10 rad/s, then −20 dB/dec

Common mistakes

• Wrong: Reading margins from the closed-loop Bode plot. — Right: Gain and phase margins come from the open-loop (loop gain) plot.
• Wrong: Forgetting that a time delay adds phase lag ωT with no magnitude change. — Right: Include delays (sampling, computation, PWM) — they often set the real phase margin.

Applications

• Power-supply loops: Engineers measure loop gain with a network analyser (injecting a small signal) and check PM/GM on a Bode plot.
• Audio & filters: Frequency response of amplifiers, crossovers and anti-aliasing filters is specified and verified on Bode plots.
• Drive tuning: Servo drives include built-in frequency-response analysers to tune current, speed and position loops.

Key takeaways

• Bode = magnitude (dB) and phase vs log ω; factors add.
• Poles −20 dB/dec and −90°; zeros the opposite.
• PM at 0 dB crossover, GM at −180° crossover; aim PM 45–60°, GM > 6 dB.
• Loop shaping: high low-frequency gain, −20 dB/dec at crossover, roll-off above.

Going further

• Bode's gain–phase relationship: For minimum-phase systems, phase is determined by the magnitude slope: −20 dB/dec ↔ −90°. That's why crossing over at −20 dB/dec guarantees a healthy phase margin.
• Sensitivity peak M_s: The peak of |S| = 1/min|1 + L| is a better robustness measure than GM/PM alone; M_s < 2 (6 dB) guarantees GM > 6 dB and PM > 29°.
• Frequency-response measurement: Swept-sine and chirp injection with a network analyser, or FFT-based identification, measure real loops including all parasitics.