A common continuous-time first-order LTI system has transfer function

$$H(s)=\frac{K}{\tau s+1},$$

where $K$ is the steady-state gain and $\tau>0$ is the time constant.

For a unit-step input and zero initial state, the response is

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

The time constant sets the speed of the transient. After one time constant, the response has completed about $63%$ of the change toward its final value; after several time constants the exponential transient is small.

The pole is

$$s=-\frac1\tau,$$

so a smaller positive $\tau$ places the pole farther left in the $s$-plane and produces faster decay.

First-order models describe many systems with one dominant storage mechanism, including thermal, electrical, fluid and actuator dynamics.