Series RLC Circuit without Source

A source-free RLC circuit consists of a resistor, inductor, and capacitor connected in series without an external energy source. The initial energy stored in the capacitor and inductor drives the circuit's behavior. The resistor dissipates this energy over time, causing the circuit response to decay exponentially.

A source-free RLC circuit requires a second-order differential equation because it contains two energy-storage elements: the inductor and capacitor. These elements interact through the resistor, which dissipates energy. The equation captures how voltage and current change across all three components simultaneously.

The two roots of the characteristic equation represent the circuit's natural frequencies and determine its natural response. These roots are expressed in terms of the damping factor and resonant frequency. The natural response is a linear combination of solutions based on these roots, revealing whether the circuit is overdamped, critically damped, or underdamped.

The damping factor determines whether the circuit exhibits overdamped, critically damped, or underdamped behavior. When the damping factor exceeds the resonant frequency, the response is overdamped with distinct real roots. When equal to the resonant frequency, the response is critically damped with equal roots.

An underdamped response occurs when the damping factor is less than the resonant frequency, resulting in complex roots. This produces oscillatory behavior where the circuit response oscillates while decaying exponentially. The complex roots indicate that energy exchanges between the inductor and capacitor before being dissipated by the resistor.

The resistor dissipates the initial energy stored in the capacitor and inductor, causing the circuit response to decay exponentially. This energy dissipation is fundamental to the circuit's behavior and leads to the exponential solution of the differential equation. Without the resistor, energy would oscillate indefinitely between the inductor and capacitor.

Critical damping occurs when the damping factor equals the resonant frequency, producing equal roots in the characteristic equation. This condition represents the boundary between overdamped and underdamped responses. At critical damping, the circuit returns to equilibrium as quickly as possible without oscillating.