Step Response Time Calculator
Calculate rise time and time constant for a first-order system step response.
Rise Time (10% to 90%)
2.197 s
Rise Time (10% to 90%) vs Time Constant (tau)
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.
Exemplo Resolvido
A temperature control system with tau = 1 s, gain = 5, step input = 1.
- 01Final value = 5 × 1 = 5
- 02Rise time = 2.197 × 1 = 2.197 s
- 03Time to 63.2% = 1 s (output = 3.16)
- 04Time to 95% = 3 s (output = 4.75)
- 05Time to 99% = 4.605 s (output = 4.95)
Perguntas Frequentes
Why is the time constant so important?
The time constant determines everything about the first-order step response. A system with tau = 0.1 s is ten times faster than one with tau = 1 s. All timing metrics (rise time, settling time) are proportional to tau.
Do first-order systems overshoot?
No. A true first-order system always approaches the final value monotonically from one side without overshooting. Overshoot occurs in second-order or higher-order underdamped systems.
How do I measure tau from experimental data?
Apply a step input and measure the time for the output to reach 63.2% of the final change. That time equals tau. Alternatively, draw a tangent at t=0; it intersects the final value line at t = tau.
Aprender