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Rise and settling time

How fast a system responds is described by the rise time and the settling time. For a first-order system both follow directly from the time constant, so halving the time constant gives a response twice as fast.

t_r \approx 2.2\,\tau

rise time from 10 % to 90 %

t_s \approx 4\,\tau

settling time to within 2 %

Symbols

$t_r$	rise time	s
$t_s$	settling time	s
$\tau$	time constant	s

Example

$\tau = 3$ s:

$t_r \approx 6.6$ s and $t_s \approx 12$ s.

The time constant can be read from a measured step response: the time to 63 % of the change.

Practise step response and first-order systems for free →

← First-order system · Final value theorem →

Part of Control Engineering: Step response and first-order systems.