Calculate rise time from damped or natural frequency and damping ratio, plus bandwidth, settling time, overshoot, frequency, and period.

Rise Time Calculator

Basic
Second-Order (ζ-based)
Converter
Bandwidth

Enter exactly 1 value to calculate the other (assumes an underdamped 2nd‑order unit‑step response with ζ = 0.5, using a 0–100% rise-time definition).

Rise Time Formula

For a standard underdamped second‑order system (0 < ζ < 1) responding to a unit step input, the 0–100% rise time (first time the response reaches the final value) depends on the damping ratio.

tᵣ = (π -) / (_d), = ((√(1 - ²)) / ())
  • Where tr is the rise time (0–100%, first crossing)
  • ζ is the damping ratio (dimensionless)
  • ωd is the damped natural frequency (rad/s), where ωd = ωn√(1 − ζ²)

If you assume ζ = 0.5, then (π − φ) ≈ 2.09439 and the formula simplifies to tr ≈ 2.09439 / ωd.

Rise Time Definition

Rise time is the time required for a signal to transition from one specified low value to a specified high value. In electronics, it is commonly measured from 10% to 90% of the final value, while in control-system step-response analysis (especially for underdamped 2nd‑order systems) a “0–100%” rise time is often defined as the first time the response reaches the final value.

Rise Time Example

How to calculate rise time?

  1. First, determine ωn and ζ (or use the Second-Order tab).

    For an underdamped second-order unit-step response, identify the natural frequency ωn and damping ratio ζ (0 < ζ < 1).

  2. Finally, compute the rise time from ωd and ζ.

    Compute ωd = ωn√(1 − ζ²), then use tr = (π − φ) / ωd with φ = arctan(√(1 − ζ²)/ζ). (If you assume ζ = 0.5, you can use tr ≈ 2.09439/ωd.)

FAQ

What is a rise time?

Rise time is the time it takes for a signal to transition from a specified low level to a specified high level. A common electronics definition is 10–90% of the final value, while some control-system texts define a 0–100% rise time for an underdamped second-order step response as the first time the response reaches the final value.

What is a damped frequency?

For an underdamped second-order system, the damped natural frequency (angular) is ωd = ωn√(1 − ζ²) in rad/s. The corresponding frequency in Hz is fd = ωd/(2π).