the response of a switched circuit

Flip a switch in a circuit and something has to happen: a capacitor cannot charge instantly, an inductor resists a sudden change in current, so right after the switch there is a brief flurry — a surge, a ring, a settling — before the circuit reaches its new steady state. The switched-circuit response is that whole story of what the currents and voltages do from the moment of switching onward, and the Laplace transform is the natural tool to compute it.

Model the circuit with its differential equation (for an RLC loop, L q'' + R q' + (1/C) q = E(t)), and represent the switching event with a Heaviside step: a battery switched on at t = 0 makes the forcing E(t) = V u(t). Transform the equation, absorbing the initial capacitor charge and inductor current as the built-in initial conditions, solve the resulting algebraic equation for Q(s) or I(s), and invert. The answer splits naturally into two parts: a transient (decaying terms set by the circuit's poles — the brief ring after the switch) plus a steady state (the constant or sinusoid the circuit relaxes toward). Step functions let you handle a switch that opens later, or a source that toggles, with no extra theory.

This is one of the showcase applications of the discontinuous-forcing toolkit. It explains the spark when you unplug something (an inductor fighting the sudden current cutoff), the inrush current when power switches on, and the design of snubbers and filters. The same method — step-function forcing, transform, transient-plus-steady decomposition — covers any sudden change applied to a linear circuit.