Damped LCR oscillation, damped frequency

Electromagnetic oscillation is the periodic exchange of energy between the electric field of a capacitor and the magnetic field of an inductor, with charge and current varying sinusoidally in time.

Differential equation

A charged capacitor C is discharged through L and R in series.

  +q |----|-q
     C    L ~~~ R
     |_________|

Kirchhoff's voltage law: L di/dt + Ri + q/C = 0 with i = dq/dt:

L d²q/dt² + R dq/dt + q/C = 0
d²q/dt² + 2γ dq/dt + ω0² q = 0 ,   γ = R/2L ,  ω0² = 1/LC

Solution

1. Try q = q0 e^(αt). Auxiliary equation: α² + 2γα + ω0² = 0, α = −γ ± √(γ² − ω0²).
2. For weak damping (γ < ω0, i.e. R² < 4L/C), write √(γ² − ω0²) = ± iω′ with

ω′ = √(ω0² − γ²) = √(1/LC − R²/4L²)

3. Solution: q = q0 e^(−Rt/2L) cos(ω′t + φ).

Frequency and significance

f′ = ω′/2π = (1/2π) √(1/LC − R²/4L²)

• The charge oscillates with the lower frequency ω′ < ω0 and with exponentially decaying amplitude q0 e^(−Rt/2L).
• Energy is dissipated as heat in R, so oscillations die out; time constant for amplitude = 2L/R.
• If R² = 4L/C (critical damping), ω′ = 0: no oscillation, charge decays fastest without overshoot.
• If R² > 4L/C (overdamped), no oscillation and slow decay.
• ω′ tells how a real LC circuit (e.g. a tuned circuit) differs from the ideal one; for small R, ω′ ≈ ω0.