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First-Order System

A first-order system has the transfer function $K/(\tau s+1)$ , with a single pole at $s=-1/\tau$ . The response to a unit step goes exponentially toward the final value $K$ , and after one time constant $\tau$ it has covered 63% of the way. This simple form shows up everywhere: temperature processes, RC circuits and simple motors.

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

first-order transfer function

G(0)=K

final value for a unit step

Symbols

$K$	static gain (final value)
$\tau$	time constant	s

Example

$G(s)=\dfrac{4}{2s+1}$ :

Final value $G(0)=4$ . Time constant $\tau=2$ s.

The pole is at

s=-1/\tau

– read

\tau

directly from the denominator's form

\tau s+1

Practise laplace and transfer functions for free →

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Part of Control Engineering: Laplace and transfer functions.