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First-order system

A first-order system responds to a step with a smooth exponential transition and no overshoot. Examples are the temperature in a tank or the voltage across a capacitor. Two numbers describe it: the gain K and the time constant τ.

G(s) = \frac{K}{\tau s + 1}

transfer function

y(t) = K\left(1 - e^{-t/\tau}\right)

response to a unit step

Symbols

$K$	steady-state gain
$\tau$	time constant	s
$t$	time	s

Example

$K = 2$ and $\tau = 5$ s. After 5 s:

$y = 2(1 - e^{-1}) \approx 1.26$ , which is 63 % of the final value 2.

After one time constant the system is 63 % of the way there. After four it has practically arrived.

Practise step response and first-order systems for free →

← Dead Time and Phase · Rise and settling time →

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