Resonance Curve Plot Generator

1. Enter the centre frequency f₀ — the resonant (natural) frequency of your system in hertz.
2. Set the quality factor Q with the slider or by typing an exact value. The damping ratio ζ = 1/(2Q) field stays linked, so editing either one updates the other.
3. Read the results panel: peak amplitude, the exact peak frequency, the −3 dB bandwidth BW = f₀/Q, and the two half-power frequencies f₁ and f₂.
4. Choose the axes — log or linear frequency, and a decibel or linear-gain vertical axis — to view the curve the way your field prefers.
5. Click “Add this curve to overlay” to freeze the current f₀/Q pair and compare several resonances on one plot. The legend lists each frozen curve; “Clear overlay” removes them.

Peak amplitude and peak frequency. For the displacement response the magnitude does not peak exactly at f₀. It peaks at the damped resonance f_peak = f₀·√(1 − 1/(2Q²)), and only when Q > 1/√2 ≈ 0.707; the peak value there is A_peak = Q / √(1 − 1/(4Q²)), which for high Q is very close to Q itself. Below Q = 0.707 the displacement response has no peak above its low-frequency (DC) value — it simply rolls off. (Critical damping is Q = 0.5 / ζ = 1; below that it is overdamped, but the peak already disappears at Q = 0.707.)

Bandwidth and the half-power points. The −3 dB (half-power) bandwidth is BW = f₀ / Q. The two half-power frequencies straddle f₀ and, for the band-pass / normalized response, are f_1,2 = f₀·(√(1 + 1/(4Q²)) ∓ 1/(2Q)); their difference is exactly BW and their geometric mean is exactly f₀. In the normalized-response mode a dashed cyan line marks the −3 dB half-power level. In the default displacement mode the orange dots and dashed guides still mark f₁ and f₂, but the displacement curve crosses them at slightly different levels, so no single −3 dB line is drawn.

Q and ζ are two views of the same thing. A high Q means low damping: a tall, narrow peak that rings for a long time. A low Q means heavy damping: a short, broad hump. The link is ζ = 1/(2Q), so Q = 10 is ζ = 0.05, and the “maximally flat” Butterworth case Q = 0.707 is ζ = 0.707 — the boundary between a peaked and a peak-free response.

A linear second-order resonator is the universal model shared by a series RLC circuit, a mass on a spring with a viscous damper, and a resonant acoustic cavity. Driven by a sinusoid of frequency f, its steady-state amplitude follows the resonance / amplitude response

A(f) = 1 / √( (1 − (f/f₀)²)² + ( (f/f₀) / Q )² )

where f₀ is the undamped natural frequency and Q is the quality factor. This is exactly the magnitude of the normalized second-order transfer function H(s) = ω₀² / (s² + (ω₀/Q)s + ω₀²) evaluated on the imaginary axis s = jω, with ω₀ = 2πf₀. The same denominator written with the damping ratio uses ζ = 1/(2Q), so the “(f/f₀)/Q” term is “2ζ(f/f₀)” — identical algebra, two notations. (The driven-resonance amplitude A(f) here is the steady-state response to a sinusoidal drive; it is the same resonator the spring-mass resonance calculator uses for its transmissibility, just expressed as the amplitude magnitude rather than the force-transmissibility ratio.)

Three quantities are then exact for this model. The peak sits at f_peak = f₀·√(1 − 1/(2Q²)) with value Q / √(1 − 1/(4Q²)) whenever Q > 0.707; otherwise there is no peak. The half-power bandwidth is BW = f₀/Q — the width between the two points where the power has fallen to half (amplitude to 1/√2, i.e. −3.01 dB). The half-power frequencies are f_1,2 = f₀·(√(1 + 1/(4Q²)) ∓ 1/(2Q)), which differ by exactly BW and whose geometric mean is exactly f₀. For high Q (Q > 10) these reduce to the familiar approximations f_1,2 ≈ f₀ ∓ BW/2.

The plot samples A(f) at a fixed, bounded number of points across the chosen log or linear frequency span, so a large or small entry can never make the page do unbounded work. All readouts are computed in double precision and written into the page as plain text. Nothing about your inputs leaves the browser.