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Damped vs forced LCR oscillation

Short answer

Damped LCR has no external source, decays; forced LCR driven by AC source E0 cos ωt oscillates at driving frequency with resonance. The differential equation is L d²q/dt² + R dq/dt + q/C = E0 cos ωt. It is mathematically identical to the driven mechanical oscillator m x'' + b x' + kx = F0 cos pt, with q↔x, L↔m, R↔b, 1/C↔k.

Compare the damped and forced LCR oscillation. Derive the differential equation of forced em-oscillation and compare it with driven mechanical oscillation.

Damped and forced LCR oscillation

Damped LCRForced LCR
No external source; charged capacitor discharges through L and RAC source E0 cos ωt drives the circuit
Equation: L q'' + R q' + q/C = 0L q'' + R q' + q/C = E0 cos ωt
Oscillates at its own frequency ω′ = √(1/LC − R²/4L²)Oscillates at the driving frequency ω
Amplitude decays as e^(−Rt/2L)Steady amplitude (after transient dies)
Energy is lost as heat, not suppliedSource supplies the energy lost in R
No resonanceResonance when ω = 1/√(LC)

Differential equation of forced e.m. oscillation

 ~E0 cos ωt ---- L ---- R ---- C ---+
   |_________________________________|

Voltage drops: L di/dt (inductor), Ri (resistor), q/C (capacitor). Kirchhoff's voltage law:

L di/dt + R i + q/C = E0 cos ωt
L d²q/dt² + R dq/dt + q/C = E0 cos ωt
d²q/dt² + 2γ dq/dt + ω0² q = (E0/L) cos ωt ,   γ = R/2L , ω0² = 1/LC

Comparison with driven mechanical oscillation

The driven mechanical equation is m x'' + b x' + kx = F0 cos pt.

MechanicalElectrical
xq
v = x'i = q'
mL
bR
k1/C
F0 cos ptE0 cos ωt
Amplitude resonance p² = ω0² − 2γ²Charge resonance ω² = 1/LC − R²/2L²
Velocity resonance p = ω0Current resonance ω = 1/√(LC)

The equations are mathematically identical, so the results of one carry over to the other.

✓ Answer checked by Claude Opus 5.5 (high effort)

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