---
title: "Sharpness of resonance and quality factor"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:sharpness-resonance-quality-factor"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/sharpness-resonance-quality-factor
site: BCT Engineering
---

# Sharpness of resonance and quality factor

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2074 Chaitra, 5 marks
- Video: [Resonance and Q Factor in Series RLC AC Circuits](https://www.youtube.com/watch?v=AUirtqrm-o0) — Engineers Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/sharpness-resonance-quality-factor
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Define the terms sharpness of resonance and quality factor. Derive the relation of quality factor in terms of band width.

## Short answer

Sharpness of resonance indicates how rapidly the amplitude drops when frequency deviates from resonance. Quality factor Q = 2π×(energy stored)/(energy lost per cycle) = ω₀/(2γ). For forced oscillations, half-power bandwidth Δω = 2γ, yielding Q = ω₀/Δω = f₀/(f₂−f₁). High Q gives sharp, narrow resonance.

## Answer

#### Sharpness of resonance
It is the rate at which the response (amplitude or current) falls as the driving frequency moves away from the resonant frequency. A sharp resonance has a tall narrow peak; it occurs for small damping. It is measured by the quality factor Q, or by the inverse of the bandwidth.

#### Quality factor
Q is defined as 2π times the ratio of the energy stored to the energy lost per cycle:
```text
Q = 2π × (energy stored) / (energy lost per cycle) = ω0 / 2γ
```
(For a damped oscillator, amplitude ∝ e^(−γt), energy ∝ e^(−2γt); energy lost in one period T ≈ E·2γT, so Q = 2π/(2γT) = ω0/2γ.) For a series LCR circuit, Q = ω0L/R = (1/R)√(L/C).

#### Q in terms of bandwidth
For forced oscillations the response (power or amplitude²) is proportional to
```text
1 / [(ω0² − p²)² + 4γ²p²]
```
1. At resonance (p ≈ ω0) the denominator is 4γ²ω0².
2. The half-power points are where the denominator is doubled, 8γ²ω0².
3. Near resonance, ω0² − p² = (ω0 − p)(ω0 + p) ≈ 2ω0(ω0 − p) and 4γ²p² ≈ 4γ²ω0²:
```text
4ω0²(ω0 − p)² + 4γ²ω0² = 8γ²ω0²   →   (ω0 − p)² = γ²   →   p = ω0 ± γ
```
4. The half-power bandwidth is Δω = p₂ − p₁ = 2γ.
5. Therefore
```text
Q = ω0/2γ = ω0/Δω = f0/(f2 − f1)
```
A large Q means a small bandwidth and a sharp resonance; a small Q means a broad, flat resonance.

## Check yourself

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

1. Sharp resonance occurs for?
   A. Small damping
   B. Large damping
   C. Large R
   D. Zero driving
   Answer: A. Small damping gives tall narrow peak. (Key point: Sharpness)

2. Q is?
   A. w0 x 2gamma
   B. 2pi x energy stored / energy lost per cycle = w0/2gamma
   C. gamma/w0
   D. Energy lost per cycle
   Answer: B. Standard definition. (Key point: Quality factor)

3. The half-power bandwidth is?
   A. gamma
   B. w0
   C. 2 gamma
   D. 4 gamma
   Answer: C. Half-power points at p = w0 +/- gamma. (Key point: Half-power bandwidth)

4. Hence Q =?
   A. Delta w/w0
   B. w0 Delta w
   C. 1/w0
   D. w0/Delta w
   Answer: D. Q = w0/2gamma = w0/Delta w; high Q means narrow bandwidth. (Key point: Q and bandwidth)

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