Step Response Calculator: Overshoot, Rise Time, Peak Time and Settling Time of First and Second Order Systems

How a system responds to a sudden step input, with the standard measures.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose first or second order.

For second order enter the natural frequency and the damping ratio. For first order enter the time constant.

The calculator gives the overshoot, the times, the output at a time you choose, and draws the response.

Formulas used

  • M_p = e^{-\pi\zeta/\sqrt{1 - \zeta^2}}
  • t_p = \frac{\pi}{\omega_n\sqrt{1 - \zeta^2}}
  • t_s\approx\frac{4}{\zeta\omega_n}
  • y = 1 - e^{-t/\tau}

Worked example

With the default numbers in the calculator above:

  1. Type of response for that damping ratio: 0.4, underdamped: it overshoots and rings
  2. Final value for the step: 1
  3. Percentage overshoot: 25.383 % (the peak is 1.25383)
  4. Peak time: 0.34278 s
  5. Damped natural frequency: 9.1652 rad/s (1.4587 Hz)
  6. Rise time from 10 to 90 percent of the final value: 0.14635 s
  7. Settling time to within 2 percent (found from the response): 0.84075 s (the rule 4/(zeta wn) gives 1 s for a lightly damped system)
  8. Output at the time you gave: 1.07609 at t = 0.5 s

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: System First order (time constant).

  1. Final value for the step: 1
  2. Time to reach 63.2 percent of the final value (one time constant): 0.5 s
  3. Rise time (10 to 90 percent): 1.0986 s
  4. Settling time to within 2 percent: 1.956 s (about four time constants)
  5. No overshoot: 0 %: a first order system never overshoots
  6. Output at the time you gave: 0.632121 at t = 0.5 s

Common questions

What is the damping ratio?

A number that says how quickly oscillations die away. Below 1 the system overshoots, at 1 it is critically damped, above 1 it is overdamped.

What is settling time?

The time for the output to stay within 2 percent of its final value.

What is a time constant?

For a first order system, the time to reach 63.2 percent of the final value. After four time constants it is within 2 percent.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any control systems text (time response of first and second order systems).

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