Derivation of RLC Resonance Formulas | Bandwidth, Quality Factor (Q_0) & Dynamic Impedance (Z_0)

In this lecture, we cover the complete, step-by-step mathematical derivations of all core formulas related to Series and Parallel Resonance in RLC Circuits. This video is specifically designed for Polytechnic Diploma, B.Tech/B.E., and Electrical/Electronics Engineering students preparing for university exams and competitive tests. ━━━━━━━━━━━━━━━━━━━━━ 📌 FORMULAS DERIVED IN THIS VIDEO: 1️⃣ Bandwidth (Series RLC): Δf = R / (2πL) or Δω = R / L 2️⃣ Dynamic Impedance (Parallel RLC Tank Circuit): Z0 = L / (CR) and I0 = V / Z0 = V / (L/CR) 3️⃣ Quality Factor (in terms of circuit parameters): Q0 = (1 / R) · √(L / C) 4️⃣ Quality Factor (in terms of selectivity & bandwidth): Q0 = ω0 / Δω = f0 / Δf ━━━━━━━━━━━━━━━━━━━━━ 📚 TOPICS & CONCEPTS COVERED: • Difference between Acceptor (Series) and Rejector (Parallel) circuits • Half-power frequencies (lower cutoff f1, upper cutoff f2) • Dynamic admittance and rationalization method for parallel tank circuit • Maximum impedance and minimum current conditions at anti-resonance • Selectivity, sharpness of resonance, and bandwidth requirements ━━━━━━━━━━━━━━━━━━━━━ ⏱️ CHAPTERS: 00:00 - Introduction & Video Overview 00:37 - Recap: Series vs Parallel Resonance Curves 04:52 - Derivation 1: Bandwidth (Δf = R/2πL) 14:08 - Derivation 2: Parallel Resonance Dynamic Impedance (Z0 = L/CR) 20:00 - Derivation 3: Quality Factor Q0 = (1/R)√(L/C) 21:36 - Derivation 4: Relation Between Q0, f0, and Δf 28:22 - Summary & Final Recap ━━━━━━━━━━━━━━━━━━━━━ 👍 Don't forget to LIKE, SHARE, and SUBSCRIBE for more Diploma & Engineering lectures! 💬 Drop your questions and feedback in the comments below. 00:00 - Introduction & Video Overview 00:37 - Recap: Series (Acceptor) vs Parallel (Rejector) Resonance Curves 03:31 - Common Resonant Circuit Characteristics & Requirements 04:52 - Derivation 1: Bandwidth of Series RLC Circuit (Δf = R/2πL) 06:15 - Half-Power Frequencies & Impedance Condition (Z = √2 R) 08:38 - Finding Resonant Frequency Condition (ω1 · ω2 = 1/LC) 11:00 - Proving Bandwidth Δω = R/L and Δf = R/2πL 14:08 - Derivation 2: Parallel Tank Dynamic Impedance (Z0 = L/CR) 15:00 - Admittance Method (YL, YC) & Complex Rationalization 16:49 - Resonance Condition (Unity Power Factor / Im{Y} = 0) 18:25 - Dynamic Admittance & Maximum Impedance Z0 = L/CR 20:00 - Derivation 3: Quality Factor Expression Q0 = (1/R)√(L/C) 21:36 - Derivation 4: Relationship Between Q-Factor & Bandwidth (Q0 = f0/Δf) 28:22 - Master Recap of All 4 Resonant Formulas 29:55 - Outro & Preview of Next Lecture (Tuning Circuits)