Step Response Time Calculator

Calculate rise time and time constant for a first-order system step response.

s

Rise Time (10% to 90%)

2.197 s

Time to 63.2% of Final Value1.000 s
Time to 95% of Final Value3.000 s
Time to 99% of Final Value4.605 s
Steady-State Output5.000

Rise Time (10% to 90%) vs Time Constant (tau)

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Formula

## First-Order Step Response A first-order system responds to a step input as an exponential approach to the final value. The speed of response is governed by the time constant tau. ### Key Relationships **y(t) = K × u × (1 - e^(-t/tau))** - At t = tau: output reaches 63.2% of final value - Rise time (10-90%): t_r = 2.197 × tau - Time to 95%: t = 3 × tau - Time to 99%: t = 4.605 × tau First-order systems have no overshoot; the output monotonically approaches the steady state.

Esempio Risolto

A temperature control system with tau = 1 s, gain = 5, step input = 1.

  1. 01Final value = 5 × 1 = 5
  2. 02Rise time = 2.197 × 1 = 2.197 s
  3. 03Time to 63.2% = 1 s (output = 3.16)
  4. 04Time to 95% = 3 s (output = 4.75)
  5. 05Time to 99% = 4.605 s (output = 4.95)