Control Theory #44 — Sinusoidal Steady-State and Frequency Response (Lecture 5 · Part 1)

Instructor: Dr. Süleyman Burak ÇELİK

We change the lens. So far we have studied LTI systems through their step response — rise time, peak time, settling time. From this lecture forward we send pure sinusoids in and ask what comes out. The answer is the frequency response — the magnitude and phase that the system applies to each frequency — and it is the foundation of every classical control design tool that follows: Bode plots, Nyquist stability, gain and phase margins, lead-lag compensators, PID tuning. Topics covered: - Why sinusoids matter — Fourier (every signal is a sum of sines), measurement (lab sweeps), and design (all classical tools live in frequency space) - Linearity promise — same frequency in, same frequency out, only amplitude scaled and phase shifted - The main result: for input sin(ωt), steady-state output = |G(jω)| · sin(ωt + ∠G(jω)) - Quick sketch of where the formula comes from (Laplace + partial fractions, transients decay) - How to compute G(jω): substitute s = jω in G(s), then extract magnitude and phase from the resulting complex number - Magnitude rule: |G(jω)| = √(A² + B²) where G(jω) = A + jB - Phase rule: ∠G(jω) = arctan(B/A); for rational G, magnitude is product of factor magnitudes, phase is sum of factor phases - Worked example: G(s) = 1/(s+1) — at ω = 0 (DC), magnitude 1 and phase 0; at ω = 1 (corner), magnitude 1/√2 ≈ 0.707 and phase −45°; at ω = 10, magnitude ≈ 0.10 and phase ≈ −84°. Low-pass filter signature. - Decibels: |G|_dB = 20 · log₁₀ |G(jω)|. Compresses the range, turns products into sums. The famous −3 dB cutoff (where |G| = 1/√2) defines bandwidth. - The full picture — magnitude and phase plotted against frequency on log axes — is a complete fingerprint of any LTI system. Big idea: the frequency response is the system, in another language. Master it and the rest of the course is geometry. Next lecture introduces the Bode plot — the efficient way to draw magnitude and phase from a transfer function. Every step shown clearly — no shortcuts. Topics: 00:00 Cover 00:03 Recap — last time we did time-domain step response 00:27 Motivation — why sinusoids matter 01:17 Thought experiment — sinusoid in, what comes out? (animated demo) 02:10 Main result — y_ss(t) = |G(jω)| · sin(ωt + ∠G(jω)) 03:19 Computing G(jω) — substitute s = jω + extract magnitude/phase 04:28 Worked example — G(s) = 1/(s+1) at ω = 0, 1, 10 06:04 Decibels — log scale + −3 dB cutoff 07:04 Full picture — magnitude and phase vs ω 07:54 Summary Playlist: Control Theory - AcEdumy GitHub: https://github.com/acedumy