Bode plots · controls

Bode unrolls the s-plane into two log-log graphs of how the system treats each frequency individually: how much it amplifies (magnitude) and how much it delays (phase). Drag a pole — watch a corner appear at $o m e g a = ∣ p ∣$ , the slope drop by 20 dB/dec, and the phase lose another 90°. Hover the Bode plot to scrub a specific frequency.

Drag the s-plane, watch the Bode plot

show asymptotes off

peak gain —

peak ω —

bandwidth (−3 dB) —

poles —

what to try

Hover the Bode plot — a vertical scrubber annotates $∣ G (j o m e g a) ∣$ and $an g l e G (j o m e g a)$ at that frequency. Find where magnitude is 0 dB.
Slide a pole leftward (more negative real part). The corner moves to higher $o m e g a$ . Slower decay = lower bandwidth.
Drag the conjugate pair toward the imaginary axis. The resonance peak grows — at $z e t a o 0$ it becomes infinite.
Toggle asymptotes. Each pole contributes one corner and a 90° phase lag; each zero gives a 90° lead. Bode = atoms summed.

show the math

Substitute $s = j o m e g a$ into the transfer function; the magnitude is

|G(jomega)| = rac{|N(jomega)|}{|D(jomega)|}, quad ext{(dB)} = 20log_{10}|G(jomega)|

and the phase is

an g l e G (j o m e g a) = a r g N (j o m e g a) - a r g D (j o m e g a)

Each real pole at $- a$ contributes a factor $rac{1}{1 + jomega/a}$ — flat below $o m e g a = a$ , rolling off at −20 dB/dec above. Each complex pair contributes −40 dB/dec.