Analyze system response over a chosen frequency span. Reveal resonance, bandwidth, and damping trends clearly. Export clean tables for lab notes and design decisions.
This calculator models a driven, damped second-order system, commonly used for oscillators and resonant sensors. The frequency response (transfer function) evaluated on the imaginary axis is:
Magnitude and phase are computed from the real and imaginary denominator parts:
When inputs are given in Hz, ω = 2πf and ω₀ = 2πf₀. A quality factor estimate is Q ≈ 1/(2ζ).
Sample inputs: K = 1.0, f0 = 5 Hz, ζ = 0.20, sweep 0.1–20 Hz.
| # | Frequency (Hz) | |H(jω)| | |H(jω)| (dB) | Phase (deg) |
|---|---|---|---|---|
| 1 | 0.10 | 1.000 | 0.000 | -0.046 |
| 2 | 2.00 | 1.190 | 1.514 | -9.739 |
| 3 | 5.00 | 2.500 | 7.959 | -90.000 |
| 4 | 7.00 | 0.758 | -2.408 | -149.135 |
| 5 | 12.00 | 0.220 | -13.151 | -171.108 |
Values are illustrative; your results depend on sweep settings and units.
Frequency response shows how output amplitude and phase vary with frequency. This calculator evaluates a second-order transfer response on the jω axis and generates a sweep table you can export for plotting. It then summarizes peak response, −3 dB bandwidth, and an estimated quality factor.
Each row is one frequency sample between your start and end values. Magnitude is steady-state gain, and phase is the lag in degrees. More points give better resolution near steep changes and help identify narrow peaks.
With low damping, the response peaks near the natural frequency. The calculator reports the peak magnitude and its frequency. Peaks can indicate sensitivity in resonant sensors, but they also amplify disturbances and measurement noise.
Bandwidth is estimated at the half-power points where magnitude drops to |H|/√2 of the peak (about −3 dB). The two cutoff frequencies define a practical operating range for filters and dynamic components. When the sweep range does not include both crossings, the tool cannot report bandwidth.
A common estimate is Q ≈ 1/(2ζ). Higher Q means a narrower bandwidth and a sharper peak. Mechanically, that implies lower energy loss per cycle; electrically, it indicates reduced resistive damping.
Phase typically moves from near 0° at low frequency toward −180° at high frequency. Near resonance, many second-order systems pass close to −90°. This matters in feedback systems because phase lag reduces stability margin and can promote oscillations.
Use Hz or rad/s; the tool converts with ω = 2πf. Sweep wide enough to bracket the expected natural frequency by at least 2× on each side. If no peak appears, widen the range, increase points, or increase K to improve visibility.
The same model supports mass–spring–damper motion, RLC circuits, vibration isolators, MEMS resonators, and simplified control-loop dynamics. Changing K, f₀, and ζ quickly shows how design choices shift peak and bandwidth.
Start with a coarse sweep to locate resonance, then refine with more points for accurate cutoffs. Compare peak magnitude and bandwidth across scenarios, export CSV for plotting, and export PDF for reporting a consistent snapshot. For design studies, record the same sweep range for every case to keep comparisons fair.
1) What does gain K change?
K scales the magnitude curve at all frequencies. It does not change the peak frequency or bandwidth for the same f₀ and ζ.
2) Why can the peak occur near, not exactly at, f₀?
Damping alters where the maximum magnitude occurs. As ζ increases, the peak broadens and can shift slightly. With high damping, the peak may disappear entirely.
3) What if bandwidth is not found?
The sweep must include both half‑power crossings. Increase the frequency span or increase the number of points. If the response is very flat (high damping), half‑power cutoffs may be ill‑defined.
4) How many sweep points should I use?
Use 80–150 for a quick scan. For sharp resonances (high Q), 200–500 points better captures the peak and cutoff frequencies.
5) Can I make a Bode plot from the output?
Yes. Export the CSV and plot magnitude and phase versus frequency. For classic Bode style, use a logarithmic frequency axis and convert magnitude to dB if needed.
6) What damping values are typical?
Light resonators may use ζ ≈ 0.005–0.05. Many mechanical systems fall around 0.05–0.3. Values above 0.5 are heavily damped and rarely show a strong resonance peak.
7) Does this handle zeros or higher‑order models?
This calculator uses a standard second‑order form with gain, natural frequency, and damping. If your system has zeros or extra poles, the same workflow applies, but the curves and metrics can differ.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.