Bode plot

A Bode plot is a system's frequency response laid out as two graphs stacked together: one showing how much the system amplifies or attenuates each frequency (gain, in decibels), the other showing how much it delays each frequency (phase, in degrees), with frequency on a logarithmic axis. Think of sweeping a tone generator from very low to very high through the system and recording, at every pitch, how loud the output is and how much it lags. That sweep is the Bode plot, and it reveals a system's character — where it acts fast, where it sags, where it filters.

Engineers love it because complicated transfer functions become simple straight-line segments on a log scale: each pole bends the gain down by 20 dB per decade and the phase down by 90°, each zero bends them back up, so you can sketch the whole curve by hand from the pole-zero locations. More importantly, the Bode plot reads out stability margins directly. Find the frequency where gain crosses 0 dB and read the phase there — how far it is above −180° is your phase margin. Find where phase hits −180° and read the gain — how far below 0 dB is your gain margin. These two numbers tell you how much extra delay or gain the loop can take before it tips into oscillation.