---
title: "Damped LCR oscillation, damped frequency"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:damped-lcr-frequency-significance"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/damped-lcr-frequency-significance
site: BCT Engineering
---

# Damped LCR oscillation, damped frequency

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2069 Asar, 5 marks; 2067 Asar, 5 marks (asked 2 times)
- Video: [Physics 47  Inductance (8 of 8) The L-R-C Circuit With Dampening Effect](https://www.youtube.com/watch?v=umRSobSlq70) — Michel van Biezen
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/damped-lcr-frequency-significance
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

What is electromagnetic oscillation? Derive the differential equation of damped LCR oscillation, solve it to find the damped frequency of the oscillation and explain its significance.

## Short answer

Electromagnetic oscillation is energy exchange between L and C. Damped LCR circuit: L d²q/dt² + R dq/dt + q/C = 0 gives d²q/dt² + 2γ dq/dt + ω0² q = 0 with γ = R/(2L). For weak damping, q = q0 e^(−γ t) cos(ω′t), where damped frequency ω′ = √(1/LC − R²/(4L²)) is lower than ω0 due to energy dissipation in R.

## Answer

**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.
```text
  +q |----|-q
     C    L ~~~ R
     |_________|
```
Kirchhoff's voltage law: L di/dt + Ri + q/C = 0 with i = dq/dt:
```text
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
```text
ω′ = √(ω0² − γ²) = √(1/LC − R²/4L²)
```
3. Solution: q = q0 e^(−Rt/2L) cos(ω′t + φ).

#### Frequency and significance
```text
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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The differential equation of damped LCR oscillation is?
   A. L q'' + R q' + q/C = E0 cos wt
   B. L q'' + q/C = 0
   C. L q'' - R q' + q/C = 0
   D. L q'' + R q' + q/C = 0
   Answer: D. Kirchhoff's law with no source. (Key point: Equation)

2. The damped angular frequency is?
   A. sqrt(1/LC - R^2/4L^2)
   B. sqrt(1/LC + R^2/4L^2)
   C. 1/sqrt(LC)
   D. R/2L
   Answer: A. w' = sqrt(w0^2 - gamma^2) with gamma = R/2L. (Key point: Damped frequency)

3. The charge amplitude decays as?
   A. e^(-Rt/L)
   B. e^(-Rt/2L)
   C. e^(-t/RC)
   D. e^(+Rt/2L)
   Answer: B. q = q0 e^(-Rt/2L) cos(w't + phi). (Key point: Amplitude decay)

4. What happens when R^2 = 4L/C?
   A. Oscillation at w0
   B. Slow oscillation
   C. No oscillation; charge decays fastest without overshoot
   D. Oscillation never decays
   Answer: C. w' = 0 gives critical damping. (Key point: Critical damping)

## Related in Engineering Physics

- [Damped vs forced LCR oscillation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-lcr-differential-compare) — Engineering Physics, chapter 1
- [Forced LCR oscillation and resonance](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-lcr-resonance-frequency) — Engineering Physics, chapter 1, asked 2×
- [LC oscillation differential equation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Alc-oscillation-differential-equation) — Engineering Physics, chapter 1
- [Damped oscillation, mechanical vs e.m.](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-mechanical-versus-electromagnetic) — Engineering Physics, chapter 1
- [Damped oscillation energy decreases with time](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-oscillation-damped-energy-decrease) — Engineering Physics, chapter 1
- [Fractional change in damped frequency](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-fractional-frequency-change) — Engineering Physics, chapter 1

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