Step Response Characteristics
\[\displaystyle G(s) = \frac{K}{\tau s + 1}\]
\[\displaystyle y(t) = K\!\left(1 - e^{-t/\tau}\right)\]
| Characteristic | Formula | Value | Annotation |
|---|---|---|---|
| Steady-state gain | \(y(\infty) = K\) | — | red dashed |
| Time constant τ | \(y(\tau) = K(1 - e^{-1}) \approx 0.632\,K\) | — | green marker |
| Rise time (10%→90%) | \(t_r = \tau\ln 9 \approx 2.197\,\tau\) | — | orange region |
| Settling time (2%) | \(t_s \approx 4\tau\) | — | purple dashed |
| Initial slope | \(\left.\dfrac{dy}{dt}\right|_{t=0} = \dfrac{K}{\tau}\) | — | gray tangent |