Transfer Function

Since the input and output signals are defined as $v_e(t)$ and $v_C(t)$ , respectively, the transfer function of this analog filter is given by, using voltage divider rule,

$$H(s) \eqsp \frac{V_C(s)}{V_e(s)} \eqsp \frac{\frac{1}{Cs}}{R+\frac{1}{Cs}} \eqsp \frac{1}{RCs+1} \eqsp \frac{1}{RC}\frac{1}{s+\frac{1}{RC}} \isdefs \frac{1}{\tau} \frac{1}{s+\frac{1}{\tau}}.$$

The parameter $\tau\isdef RC$ is called the RC time constant, for reasons we will soon see.