# What is the gain and time constant of a system with a constant output?

For example, if your transfer function is defined as :

$$\frac{H(s)}{Q(s)} = \frac{1}{5s}$$

What would the gain and time constant be in this case (since it's usually in the form of $\dfrac{k}{Ts+1}$ where $k$ and $T$ are the time constant and gain, respectively)?

(This is the transfer function for a tank being filled, but the exit stream is a pump so it is constant. Where $H$ and $Q$ are deviation variables for output and input, respectively)

$G(s) = \frac{1}{s}$ is called an integrator. It continuously integrates your signal and therefore it has no time constant, hence $\tau = 0$.
Your system $\frac{H(s)} {Q(s)} = \frac{1}{5s}$ can be written as an integrator multiplied with a gain of $\frac{1}{5}$.
Therefore the pole-zero gain is $k_{pz} = \frac{1}{5}$.
The static gain of your function $k_s = \lim_{s\to 0} \frac{1}{5s} = \infty$ or you might say there is no static gain.