Control Theory #46 — Bode Phase Plot (Lecture 5 · Part 3)

Instructor: Dr. Süleyman Burak ÇELİK

The second half of the Bode plot — phase. Same factoring trick as magnitude, different rules, same speed. Sum the per-atom phase asymptotes and you have the full phase curve, ready to read stability margins off. Topics covered: - Why phases add: ∠(G·H) = ∠G + ∠H; ∠(1/G) = −∠G — denominator factors flip the sign - Phase rule per atom: K → 0° (or ±180° if K‹0); integrator 1/s → constant −90°; 1/s^n → −90n°; simple pole 1/(s+a) → 0° below corner, −90° above; simple zero (s+a) → mirror; right-half-plane zero → non-minimum phase signature - Decade rule for simple poles/zeros: phase is flat at 0° for ω ‹ a/10, ramps ±45°/decade between a/10 and 10·a, flat at ±90° for ω › 10·a; phase is exactly ±45° at the corner - Worked example: G(s) = 10/[s(s+1)(s+10)] — phase goes −90° → −180° → −270° as the integrator and two simple poles add their contributions - Asymptotic vs actual: smooth curve deviates from the piecewise-linear ramp by ~6° at the kink points; complex pairs with low damping can deviate much more - Phase margin preview: at the gain-crossover frequency ω_gc (where |G(jω)|=1), the phase margin is 180° + ∠G(jω_gc) — a small margin means the closed loop teeters near instability The recipe in 10 seconds: factor → integrator/differentiator gives constant offset → simple poles/zeros give 2-decade ramps centered on each corner → complex pairs give double-speed swings. Big idea: the Bode plot is now complete — magnitude and phase, both drawn fast from atom asymptotes. Together they give the system's full frequency response, and they tell us — at every frequency — how stable a closed-loop design built around this loop gain will be. Next lecture we put magnitude and phase side-by-side on real examples and start reading. Every step shown clearly — no shortcuts. Topics: 00:00 Cover 00:03 Recap — magnitude done, today phase 00:23 Why phases add: ∠(G·H) = ∠G + ∠H 01:11 Phase rule per atom: 0/±180, ±90, 0 → ±90, 0 → ±180 02:31 Decade rule: ramp ±45°/dec across two decades 03:21 Worked example: −90 → −180 → −270 04:51 Asymptotic vs actual: ~6° at kinks 05:36 Preview — phase at gain crossover = phase margin 06:15 Summary Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy