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Sharpness of resonance and quality factor

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.

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

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:

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

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:
4ω0²(ω0 − p)² + 4γ²ω0² = 8γ²ω0²   →   (ω0 − p)² = γ²   →   p = ω0 ± γ
  1. The half-power bandwidth is Δω = p₂ − p₁ = 2γ.
  2. Therefore
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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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