--- title: "Oscillations in LC and RLC Circuits" description: "An ideal LC circuit's charge obeys the same equation as displacement in a linear mass-spring system. In a freely decaying series RLC circuit, resistance supplies the damping term and dissipates stored" canonical: https://lightmysky.com/learn/science/oscillations-in-lc-and-rlc-circuits-mt_QJsqX7xzkZ source: https://lightmysky.com/learn/science/oscillations-in-lc-and-rlc-circuits-mt_QJsqX7xzkZ.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Oscillations in LC and RLC Circuits An ideal LC circuit's charge obeys the same equation as displacement in a linear mass-spring system. In a freely decaying series RLC circuit, resistance supplies the damping term and dissipates stored energy as heat. Equivalent equations and initial conditions give corresponding solutions. Subject: Science · Area: Electricity & Magnetism · Ages 20 to 21 Page: https://lightmysky.com/learn/science/oscillations-in-lc-and-rlc-circuits-mt_QJsqX7xzkZ ## Ready when they can - Derives the ideal LC equation from the loop rule and identifies its natural angular frequency - Matches charge, current, inductance and inverse capacitance (1/C) to displacement, velocity, mass and spring constant - Describes how resistance damps the oscillation and where the energy goes ## Lesson: Charge on a spring: LC ringing and RLC damping An ideal LC loop has inductance L, capacitance C, no resistance and no source. Let q be the charge on one capacitor plate. Capacitance relates charge to voltage: q = C vC, so vC = q/C. Choose current I = dq/dt. The loop rule gives L q'' + q/C = 0, where q'' is d squared q/dt squared. Divide by L: q'' + q/(LC) = 0. Its natural angular frequency is omega0 = 1/sqrt(LC), in radians per second. Compare L q'' + q/C = 0 with m x'' + k x = 0. Charge q corresponds to displacement x, inductance L to mass m, and 1/C to spring stiffness k. These are corresponding linear equations, not identical physical quantities. Matching initial conditions gives corresponding motions. Source-free series RLC has L > 0, C > 0, R >= 0 and L q'' + R q' + q/C = 0. U = q squared/(2C) + L I squared/2 is total stored energy. dU/dt = -R I squared: energy becomes heat at rate R I squared in the resistor. R = 0 is undamped. For 0 < R < 2 sqrt(L/C), nonzero charge has envelope A exp(-Rt/(2L)), with constant amplitude A, and omega_d = sqrt(1/(LC) - (R/(2L)) squared), below omega0 and close only for weak damping. At or above critical R, motion is nonoscillatory. A capacitor's voltage is q/C. Charging work is the integral of (q/C) dq from 0 to q, giving electric energy q squared/(2C). In ideal LC, U = q squared/(2C) + L I squared/2 is constant. Start at q = Qmax, I = 0: all energy is electric. At T/4, q = 0 and current magnitude peaks, so all energy is magnetic. Charge and current have angular frequency omega0; each energy depends on a square and repeats every half cycle. For a free series RLC response with positive L and C, let Rcrit = 2 sqrt(L/C). R = 0 is undamped. Positive R below Rcrit is underdamped, R = Rcrit is critical, and R > Rcrit is overdamped. From the same nonzero capacitor charge and zero initial current, critical damping gives the fastest nonoscillatory approach to zero charge. Increasing R beyond critical slows that approach. This compares circuits with the same L and C. With the chosen reference direction, I = dq/dt. This matches velocity v = dx/dt in the spring model. If q(t) = Qmax cos(omega0 t), then I(t) = -omega0 Qmax sin(omega0 t). Current is zero at a charge extreme and has its largest magnitude when charge is zero. *(drawing: The source-free loop rule is vL + vC = 0. Substitute I = dq/dt, then divide by L to get q'' + q/(LC) = 0.)* **Recap.** Ideal LC charge oscillates at omega0 = 1/sqrt(LC). In free series RLC, resistance dissipates energy; underdamped oscillations have a lower frequency and stronger damping can prevent oscillation. *Original lesson, questions and flashcards authored for this learning dataset.* ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Self-Inductance and Energy Stored in a Magnetic Field](https://lightmysky.com/learn/science/self-inductance-and-energy-stored-in-a-magnetic-field-mt_BLyLSJT-Fr) - [Damped Oscillations and the Three Damping Regimes](https://lightmysky.com/learn/science/damped-oscillations-and-the-three-damping-regimes-mt_JCpYpFVn6s) ## Optional connections - [Damped and Driven Oscillations](https://lightmysky.com/learn/mathematics/damped-and-driven-oscillations-mt_Y8K1njb1PH) ## Opens up - [Impedance and Phasors in AC Circuits](https://lightmysky.com/learn/science/impedance-and-phasors-in-ac-circuits-mt_vQVlVWv2ln)