--- title: "Forced LCR oscillation and resonance" subject: "Engineering Physics" chapter: 1 key: "semester_1/engineering_physics:forced-lcr-resonance-frequency" source: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-lcr-resonance-frequency site: BCT Engineering --- # Forced LCR oscillation and resonance Engineering Physics · Chapter 1: Oscillation - Asked in: 2071 Shrawan, 5 marks; 2068 Chaitra, 5 marks (asked 2 times) - Video: [LCR frequency response & quality | A.C. | Physics | Khan Academy](https://www.youtube.com/watch?v=PfBQEhLKDRc) — Khan Academy India - English - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-lcr-resonance-frequency - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1 ## Question What is resonance? Formulate the differential equation of forced electromagnetic oscillation (LCR circuit with AC source). Then determine the expression for the current amplitude and resonant frequency, and explain the condition of current resonance. ## Short answer Resonance is maximum response when driving frequency equals natural frequency. Forced LCR circuit: L d²q/dt² + R dq/dt + q/C = E0 cosωt. Current amplitude I0 = E0/√[R²+(ωL-1/ωC)²] peaks at ω_r = 1/√(LC), where I0 = E0/R and current is in phase with emf. ## Answer **Resonance** is the condition in which a system driven by a periodic external force responds with maximum amplitude. It happens when the driving frequency matches the natural frequency of the system. #### Differential equation A series L, C, R circuit is driven by an AC source E = E0 cos ωt. ```text ~E0 cos ωt ---- L ---- R ---- C ---+ |_________________________________| ``` By Kirchhoff's law, L di/dt + Ri + q/C = E0 cos ωt. With i = dq/dt: ```text L d²q/dt² + R dq/dt + q/C = E0 cos ωt ``` This matches the driven mechanical oscillator m x'' + b x' + kx = F0 cos pt with m → L, b → R, k → 1/C. #### Current amplitude Steady state: try q = q0 cos(ωt − φ). Substitute, write cos ωt = cos(ωt − φ) cosφ − sin(ωt − φ) sinφ, and compare the coefficients of cos(ωt − φ) and sin(ωt − φ): ```text q0 (1/C − Lω²) = E0 cosφ q0 R ω = E0 sinφ ``` Squaring and adding: q0 = E0 / √[(1/C − Lω²)² + R²ω²]. The current is i = dq/dt = −ωq0 sin(ωt − φ), so the current amplitude is I0 = ωq0: ```text I0 = E0 / √[R² + (ωL − 1/ωC)²] ``` The current lags the emf by an angle ψ = φ − π/2, where tanψ = (ωL − 1/ωC)/R. Here X = ωL − 1/ωC is the net reactance and Z = √(R² + X²) is the impedance, so I0 = E0/Z. #### Resonant frequency and condition I0 is maximum when the denominator is minimum, i.e. when X = 0: ```text ωL = 1/ωC → ω_r = 1/√(LC) , f_r = 1/(2π√(LC)) ``` At this frequency: - Z = R (minimum), so I0(max) = E0/R. - ψ = 0: the current is in phase with the applied emf. - The circuit behaves as purely resistive; the voltages across L and C are equal and opposite. - The resonant frequency equals the natural frequency of the undamped LC circuit. A smaller R gives a taller, sharper peak. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is resonance? A. Maximum response when the driving frequency matches the natural frequency B. Zero current at a particular frequency C. Free oscillation at the natural frequency D. Decay of amplitude with time Answer: A. Resonance is the large response of a driven system at matching frequency. (Key point: Resonance) 2. The current amplitude of a driven series LCR circuit is? A. I0 = E0 / (R + wL + 1/wC) B. I0 = E0 / sqrt(R^2 + (wL - 1/wC)^2) C. I0 = E0 R D. I0 = E0 / sqrt(R^2 + (wL + 1/wC)^2) Answer: B. Squaring and adding the coefficient equations gives q0; I0 = w q0 gives E0/Z. (Key point: Current amplitude) 3. Resonant frequency occurs when? A. R = 0 B. wL = R C. wL = 1/wC, so w = 1/sqrt(LC) D. w = sqrt(R/LC) Answer: C. The reactance X = wL - 1/wC is zero, making Z minimum. (Key point: Resonant frequency) 4. Which is true at current resonance? A. Z is maximum B. Current lags emf by 90 degrees C. I0 = 0 D. Z = R, current in phase with emf, I0 = E0/R Answer: D. X = 0 gives Z = R, phi = 0, and the maximum current E0/R. 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