Step Response Time Calculator — 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.
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.
- Final value = 5 × 1 = 5
- Rise time = 2.197 × 1 = 2.197 s
- Time to 63.2% = 1 s (output = 3.16)
- Time to 95% = 3 s (output = 4.75)
- Time to 99% = 4.605 s (output = 4.95)