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.
What a learner can do afterwards
- 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
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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.
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.
Original readings and flashcards authored for this learning dataset. Video references credit Khan Academy.
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