Control Theory #45 — Bode Magnitude Plot (Lecture 5 · Part 2)

Instructor: Dr. Süleyman Burak ÇELİK

A graphical recipe to draw the magnitude curve of any rational transfer function in seconds. The trick — log frequency on the x-axis, decibels on the y-axis — turns multiplication of factors into addition of straight-line asymptotes. Master four atom types and the rest of classical control becomes geometry. Topics covered: - Why log frequency and decibels: compress 6 orders of magnitude into a manageable strip; |G·H|_dB = |G|_dB + |H|_dB turns products into sums - Factor decomposition into four atom types: constant gain K, integrator/differentiator, simple pole/zero, complex conjugate pair - Magnitude rule per atom: K → flat at 20 log K dB; 1/s → −20 dB/dec through 0 dB at ω=1; 1/(s+a) → flat below corner, −20 dB/dec above; complex pair → ±40 dB/dec + Q peak - 4-step construction recipe: factor → identify corners → start at low frequency → walk corners changing slope per atom - Worked example: G(s) = 10/[s(s+1)(s+10)] — slopes −20 → −40 → −60 dB/dec at corners ω=1 and ω=10 - Asymptotic vs actual: the actual smooth curve deviates from the piecewise-linear asymptote by ±3 dB at simple corners; complex pairs can deviate much more depending on damping The ten-second mental procedure: factor, find corners, sketch slope segments, smooth the corners by 3 dB. That is the entire Bode magnitude. Big idea: this is the first half of the Bode plot — the magnitude. Together with the phase (next lecture) it gives a complete fingerprint of the system at every frequency, ready to feed into stability analysis, gain/phase margins, lead-lag and PID design. Every step shown clearly — no shortcuts. Topics: 00:00 Cover 00:03 Recap — frequency response defined, but how to draw it fast? 00:33 Why log + dB: compress range, turn products into sums 01:22 Factor decomposition: four atom types 02:14 Atom rules: per-atom magnitude asymptotes 03:55 The 4-step construction recipe 04:36 Worked example: G(s) = 10/[s(s+1)(s+10)] 06:15 Asymptotic vs actual: ±3 dB at corners 06:55 Summary Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy