---
title: "Damped vs forced LCR oscillation"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:forced-lcr-differential-compare"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-lcr-differential-compare
site: BCT Engineering
---

# Damped vs forced LCR oscillation

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2072 Kartik, 5 marks
- Video: [SHM - PART I - Damped and Forced Oscillations](https://www.youtube.com/watch?v=-cpes3eA25Y) — PhysicsPrep 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-lcr-differential-compare
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

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

## Answer

#### Damped and forced LCR oscillation
| Damped LCR | Forced LCR |
|---|---|
| No external source; charged capacitor discharges through L and R | AC source E0 cos ωt drives the circuit |
| Equation: L q'' + R q' + q/C = 0 | L 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 supplied | Source supplies the energy lost in R |
| No resonance | Resonance when ω = 1/√(LC) |

#### Differential equation of forced e.m. oscillation
```text
 ~E0 cos ωt ---- L ---- R ---- C ---+
   |_________________________________|
```
Voltage drops: L di/dt (inductor), Ri (resistor), q/C (capacitor). Kirchhoff's voltage law:
```text
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.

| Mechanical | Electrical |
|---|---|
| x | q |
| v = x' | i = q' |
| m | L |
| b | R |
| k | 1/C |
| F0 cos pt | E0 cos ωt |
| Amplitude resonance p² = ω0² − 2γ² | Charge resonance ω² = 1/LC − R²/2L² |
| Velocity resonance p = ω0 | Current resonance ω = 1/√(LC) |

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

## Check yourself

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

1. Forced LCR obeys?
   A. L q'' + R q' + q/C = 0
   B. L q'' + R q' + q/C = E0 cos wt
   C. L q'' + q/C = E0
   D. L q'' + R q' = E0
   Answer: B. Source adds a driving term. (Key point: Forced equation)

2. Forced LCR oscillates at?
   A. w'
   B. w0 always
   C. The driving frequency w
   D. Zero
   Answer: C. After transients. (Key point: Frequency)

3. The mechanical analogue of R is?
   A. m
   B. k
   C. F0
   D. b, damping constant
   Answer: D. m -> L, b -> R, k -> 1/C. (Key point: Analogue)

4. Current resonance occurs at?
   A. w = 1/sqrt(LC)
   B. w^2 = 1/LC - R^2/2L^2
   C. w = R/L
   D. w = 0
   Answer: A. The charge resonance is the one with the damping correction. (Key point: Resonance)

## Related in Engineering Physics

- [Damped LCR oscillation, damped frequency](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adamped-lcr-frequency-significance) — Engineering Physics, chapter 1, asked 2×
- [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
- [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 energy decreases with time](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-oscillation-damped-energy-decrease) — Engineering Physics, chapter 1
- [Sharpness of resonance, forced e-m oscillation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-em-sharpness-current-amplitude) — Engineering Physics, chapter 1

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