First-Order Step Response
Time-constant milestones, rise and settling time, bandwidth and time-to-target.
—
One number sets the whole curve
A first-order system K / (1 + sτ) rises toward its final value on a single
exponential, and every milestone is a fixed multiple of the time constant τ:
That's why the rules of thumb are worth memorising: τ → 63.2 %, 2τ → 86.5 %,
3τ → 95 %, 5τ → 99.3 % ("settled"). The 10–90 % rise time is
2.197 τ, the 2 % settling time is 3.91 τ, and the −3 dB bandwidth is
1/(2πτ) — the pole sits at −1/τ. A first-order lag is the cleanest
system there is: no overshoot, no ringing, just an exponential set entirely by τ. For systems
that do ring, step up to the second-order
response.