5.7 : Second-Order Circuits

Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.

Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in an RLC circuit, initial energy stored in the capacitor and inductor initiates the circuit. Applying Kirchhoff's voltage law and performing a time derivative yields a second-order differential equation.

Its coefficients, determined by resistance, capacitance, and inductance, manifest as the damping coefficient and resonant frequency.

The damping coefficient plays a critical role in these circuits, signifying the extent of damping caused primarily by the resistor. It directly influences the pace at which energy dissipates within the system, effectively controlling the rate of energy loss. On the other hand, the resonant frequency is a key characteristic that represents the circuit's innate oscillation frequency. It measures how quickly energy is exchanged between the inductor and capacitor in the circuit, illustrating the circuit's natural tendency to oscillate at a particular frequency.

The damping coefficient dictates how fast energy is lost in the system due to resistance. At the same time, the resonant frequency highlights the circuit's inherent oscillation speed as energy shifts between the inductor and capacitor. These two factors are crucial in understanding and analyzing the behavior of second-order circuits.