Network function

Any circuit with a single input source $X(s)$ and zero initial conditions and any one of its internal or output signals $Y(s)$ can be written as:

$$Y(s)=T(s)X(s)$$

where

$$T(s)=\frac{Y(s)}{X(s)}$$ $$\text{Network function}=\frac{\text{Zero-state response}}{\text{Input signal}}$$

$T(s)$ is a rational function of $s$ called a network function. It transforms the input into some response.¹

Poles

Poles of $Y(s)$ that come from $T(s)$ are called natural poles. From $X(s)$, they’re called forced poles. If we take the inverse transform of $Y(s)$, when we roughly get:

$$y(t)=\sum k_j{e}^{p_{j}t}+k_l{e}^{p_{l}t}$$

where we have exponential terms corresponding to the natural and forced response.

A circuit is stable if its natural response dies down to zero over time (in the steady state). This is important because it means it’s fundamentally controllable. If a circuit continued to oscillate forever, an unstable circuit, it wouldn’t be very useful to us.

Footnotes

1. From Prof Najm’s lecture notes. ↩