Within the field of electrical circuits, source-free RLC circuits present an intriguing domain. These circuits comprise a series arrangement of a resistor, inductor, and capacitor, operating independently of external energy sources. Their initiation hinges upon utilizing the initial energy stored within the capacitor and inductor to instigate their functionality. Their mathematical equation, a second-order differential equation, sets these circuits apart. This equation captures how the circuit's components interact, forming the basis for understanding its behavior.

The resistor in this circuit plays a significant role by dissipating energy, leading to an exponential solution for the differential equation. Substituting this solution yields a quadratic equation, and the two roots of this equation hold special significance. These roots are the circuit's natural frequencies and are instrumental in describing its natural response.

Expressed in terms of the damping factor and resonant frequency, these roots provide insights into the circuit's behavior. If the damping factor surpasses the resonant frequency, the circuit exhibits an overdamped response with distinct real roots. When the damping factor equals the resonant frequency, a critically damped response ensues, characterized by equal roots. Finally, if the damping factor falls short of the resonant frequency, the circuit enters an underdamped state with complex roots.

Various response scenarios within source-free RLC circuits offer an intriguing and valuable aspect of circuit analysis. Further exploration of each case provides a comprehensive understanding of their behavior and practical applications in electrical circuits.