Bode Plots

A Bode plot shows a system's gain and phase versus frequency on log scales, making loop shaping and margin reading direct.

Frequency Response, Drawn

Setting s = j*omega in a transfer function gives the frequency response G(j*omega), a complex number at each frequency. A Bode plot displays its magnitude (in decibels) and phase (in degrees) against frequency on a logarithmic axis. Two curves capture how the system treats sinusoids of every frequency.

Why log scales help

On log-magnitude versus log-frequency axes, the contributions of individual poles and zeros become straight-line asymptotes that simply add. A first-order pole contributes a slope of minus 20 dB per decade above its corner frequency and about minus 90 degrees of phase; a zero contributes the opposite. Complex plots are sketched by summing these pieces.

Reading key features

• Corner (break) frequencies mark where each pole or zero begins to act.
• The gain crossover frequency, where magnitude passes 0 dB, sets closed-loop bandwidth.
• The slope through crossover indicates stability: a shallow -20 dB/decade slope is desirable.
• Phase near crossover gives the phase margin directly.

Loop shaping

Design in the Bode domain means shaping the open-loop magnitude curve: high gain at low frequency for tracking and disturbance rejection, a controlled slope through crossover for stability, and steep rolloff at high frequency to reject noise and unmodeled dynamics. Lead and lag compensators are added precisely to bend this curve where needed.

Margins from the plot

Bode plots make gain and phase margins visible. Phase margin is how far the phase sits above minus 180 degrees at gain crossover; gain margin is how far the gain sits below 0 dB at the frequency where phase crosses minus 180 degrees. Both are read straight off the curves.

Because they connect naturally to physical measurements (drive a plant with sinusoids, record amplitude and phase), Bode plots bridge theory and experiment better than most tools, and remain the default view for single-loop frequency-domain design.