A first-order system covers 63% of a step in one time constant and is effectively settled after 4–5.
Follows directly from the exponential 1 − e^(−t/τ). Knowing τ tells you the whole response timeline.
Only for a genuine first-order system. Add dead time and nothing happens at all for a while before the curve starts; make it second-order and it can overshoot and ring past five time constants. Fitting a first-order model to either gives a time constant that means nothing.
At 5 τ the rule says 0.0069 and the exact answer is 0.0067 — 2.9% high. At the point most people use it the shortcut is already outside its own 2% tolerance. It only holds between 1 τ and 3.4 τ, which is not where it gets used.
1/e is 0.3679, not 0.37, and the rounding is raised to the power of however many time constants you count. One τ is out by half a per cent; by five it is 3% and by seven 4%. The rule is fine for "is it there yet" and quietly wrong if you chain it.
The rule against the exact answer, computed across the range. Inside the shaded band the shortcut is close enough to use; outside it, reach for the calculator.
Estimate with the rule, then check it against the calculator that models it properly.
Open Mass-Spring-Damper →A first-order system covers 63% of a step in one time constant and is effectively settled after 4–5. Follows directly from the exponential 1 − e^(−t/τ).