Poles and Zeros

In our simple RC-filter example, the transfer function is

$$H(s) = \frac{\tau}{s+\tau}.$$

Thus, there is a single pole at $s=-\tau$, and we can say there is one zero at infinity as well. Since resistors and capacitors always have positive values, the time constant $\tau = 1/(RC)$ is always non-negative. This means the impulse response is always an exponential decay--never a growth. Since the pole is at $s=-\tau$, we find that it is always in the left-half $s$ plane. This turns out to be the case also for any complex analog one-pole filter. By consideration of the partial fraction expansion of any $H(s)$, it is clear that, for stability of an analog filter, all poles must lie in the left half of the complex $s$ plane. This is the analog counterpart of the requirement for digital filters that all poles lie inside the unit circle.