Step response

The step response describes the result if the input to a system is the step function, i.e.: Then we have:

$$G(s)=\frac{T(s)}{s}$$ $$G(s)=\frac{k_0}{s}+\frac{k_1}{s-p_1}+\cdots +\frac{k_n}{s-p_n}$$

where only the first term is a forced pole, and the rest are the natural poles of $T(s)$. Additionally, since we know that the transfer function $T(s)=H(s)$, the impulse response:

$$G(s)=\frac{H(s)}{s}$$ $$g(t)={\int }_0^{t}h(\tau )d\tau$$ $$h(t)=\frac{d}{dt}g(t)$$

where $g(t)$ is continuous. What this means is that we can find the impulse response of a circuit by applying $u(t)$ as an input, then differentiating its output. These relations are summarised below: