Rise Time⚠ unverified
Physics / Controls · Approximate rise time of a second-order system step response
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn | ωn | rad/s | 10.0 | Natural frequency |
| zeta | ζ | — | 0.5 | Damping ratio |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| tr | tr | s | Rise time |
The science & history
Understanding the Parameters
- $\omega_n$ — sets the time scale; $t_r \propto 1/\omega_n$.
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$\zeta$ — physically affects rise time (more damping often slows rise slightly) but is ignored by $t_r = 1.8/\omega_n$.
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$t_r$ — order-of-magnitude estimate only; definitions of rise (0–100%, 10–90%) differ.
Derivation (Approaching a Proof)
For a pure second-order prototype, analytical 10–90% rise times are tabulated vs $\zeta$. Near $\zeta \approx 0.5$–$0.7$, $t_r \omega_n$ is order-1; the constant 1.8 is a single-point fit ($\omega_n t_r \approx 1.8$). Not a first-principles identity for all $\zeta$.
History
Rise-time specifications appear in classical step-response design alongside overshoot and settling time.
Related Concepts: Settling Time 2%, Peak Overshoot, Bandwidth
Notes: Registry calculator rise-time (unverified). Unused input: $\zeta$. Approximate only.