Second-Order System Calculator: Overshoot, Rise Time, Settling Time and Bandwidth from Damping Ratio
Time and frequency response of a second-order system.
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How this calculator works
Enter the natural frequency and the damping ratio.
Enter the DC gain.
The calculator gives the response measures and a chart.
Formulas used
- G(s) = \frac{K\omega_n^2}{s^2 + 2\zeta\omega_ns + \omega_n^2}
- PO = 100\,e^{-\pi\zeta/\sqrt{1 - \zeta^2}}
- t_s \approx \frac{4}{\zeta\omega_n}
Worked example
With the default numbers in the calculator above:
- Poles of the system: -5 plus and minus 8.6603 j
- Percent overshoot: 16.303 %
- Peak time: 0.36276 s
- Rise time (10 to 90 percent): 0.1638 s
- Settling time (2 percent), from 4 / (zeta wn): 0.8 s
- Peak of the frequency response (relative to the DC gain): 1.1547 (1.249 dB) at 7.0711 rad/s
- Bandwidth (-3 dB): 12.72 rad/s
- Final value of the step response: 1
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What does the damping ratio do?
A low damping ratio gives a fast response that overshoots and rings. A ratio of 1 is the fastest response with no overshoot. A high ratio is slow.
What damping ratio gives about 5 percent overshoot?
About 0.69. A ratio of 0.707 gives about 4.3 percent.
How does bandwidth relate to speed?
A higher bandwidth means a faster rise time. For a given damping ratio the rise time is roughly inversely proportional to the natural frequency.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any control systems text (second-order system response).
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