📐 Bode Plot: Frequency Response and Stability Margins

About this simulation

A Bode plot shows how a linear system's open-loop transfer function H(s) responds to sinusoidal inputs at every frequency: a magnitude curve (in decibels) and a phase curve (in degrees), both plotted against frequency on a logarithmic scale. Here the two curves float as stacked 3D panels that share the same log-frequency axis, so you can see exactly how a change in one plot lines up with the other.

How it works

• The modelled system is H(s) = K·(1+s/ωz) / [s·(1+s/ωp1)·(1+s/ωp2)] — a gain, an integrator, two real poles, and an optional real zero.
• Gain crossover ωc — the frequency where the magnitude curve crosses 0 dB. The phase margin is read off the phase curve at that same frequency, relative to −180°.
• Phase crossover ω₁₈₀ — the frequency where the phase curve crosses −180°. The gain margin is read off the magnitude curve at that same frequency, relative to 0 dB.
• Cyan and orange vertical lines mark ωc and ω₁₈₀ on both panels; the coloured bracket shows the actual margin — green when it adds stability, red when it has gone negative (unstable).
• Cone markers on the baseline mark each pole's break frequency; the ring marks the zero. Notice how the magnitude slope steepens by −20 dB/decade at each pole and flattens by +20 dB/decade at the zero.

Controls

• Gain K — scales the whole magnitude curve up or down without touching phase.
• Pole 1 / Pole 2 (ωp1, ωp2) — break frequencies of the two real poles; each drags the phase down and steepens the magnitude roll-off.
• Zero (ωz) — break frequency of an optional real zero, which pushes phase back up; toggle it on with Enable zero.
• ↺ Reset restores the default system. Drag to rotate the 3D view, scroll to zoom.

Did you know?

Hendrik Bode developed this graphical method at Bell Labs in the 1930s to keep telephone-repeater feedback amplifiers stable. The same margins now govern everything from autopilot loops to op-amp circuits: a healthy design typically keeps phase margin above about 45° and gain margin above roughly 6–10 dB.