2.2j Second-Order Circuits

Second-Order Circuits

Second-order circuits are electrical circuits whose behavior is described by a second-order differential equation. These circuits contain both inductors and capacitors. Second-order circuits follow the general equation
$\frac{d^2}{d{t}^2}x(t)+2\alpha \frac{d}{dt}x(t)+{\omega }_0^{2}x(t)=f(t),$
where $\alpha$ is the damping coefficient, and ${\omega }_0$ is the resonant frequency of the circuit at which the impedance is purely real.

The solutions to the second-order differential circuits can be split into 3 cases:

1. Overdamped Response ($\alpha >{\omega }_0$):
   $x(t)=A_1{e}^{s_{1}t}+A_2{e}^{s_{2}t}+x(\infty ),$
   where $s=-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$ and the coefficients $A_k=$

2. Critically Damped Response ($\alpha ={\omega }_0$)

3. Underdamped Response ($\alpha <{\omega }_0$)