Series RLC Circuit with Source

Consider the operation of an automobile ignition system, a crucial component responsible for generating a spark by producing high voltage from the battery. This system can be described as a simple series RLC circuit, allowing for an in-depth analysis of its complete response.

In this context, the input DC voltage serves as a forcing step function, resulting in a forced step response that mirrors the characteristics of the input. Applying Kirchhoff's voltage law to the circuit yields a second-order differential equation. Remarkably, this equation strongly resembles the second-order differential equation characterizing a source-free RLC circuit. This similarity underscores that the presence of the DC source does not alter the fundamental form of the equations.

The complete solution to this equation comprises both transient and steady-state responses.

The transient response, which diminishes over time, aligns with the solution for source-free circuits and encompasses scenarios involving overdamped, critically damped, and underdamped behaviors. On the other hand, the steady-state response corresponds to the final value of the capacitor voltage, which equals the source voltage. The constants involved in these responses can be determined from the initial conditions of the circuit.