Second-order circuits are circuits whose behavior is governed by a second-order linear differential equation. This happens when the circuit contains two independent energy-storage elements such as two capacitors, two inductors, or one capacitor and one inductor (which is the most common case).

A second-order transient circuit is a second-order circuit analyzed during the time interval when the circuit is transitioning from one steady state to another after a change such as switching, connecting, or disconnecting a source. In second-order transient problems it is common to assume that the circuit is initially at steady-state, a perturbation occurs at time $t=0$, and we need to find the behaviur of the circuit for $t \ge 0$.

In general, the governing differential equation in a second-order circuit can be written as $$\begin{equation}\frac{d^2y(t)}{dt^2} + 2\alpha \frac{dy(t)}{dt} + \omega_0^2 y(t) = f(t)\end{equation}$$ where $y(t)$ is either a voltage or a current and $f(t)$ is a function (often refered to as forcing function or forcing term) that depends on the voltage and current sources in the circuit. Constants $\alpha$ and $\omega_0$ are called the damping factor (or Neper frequency) and the undamped natural frequency of the circuit, respectively and depend on the circuit parameters (resistances, inductances, and capacitances). The above equation is subject to two initial conditions, which are usually the value of $y(t)$ and its first derivative at $t=0$. In the case of transient circuits, these initial conditions can usually be found by analyzing the circuit at $t=0^-$ (just before the perturbation) and applying the appropriate initial conditions for capacitors and inductors.

The natural response of a second-order circuit is obtained by solving the corresponding homogeneous equation $$\begin{equation}\frac{d^2y(t)}{dt^2} + 2\alpha \frac{dy(t)}{dt} + \omega_0^2 y(t) = 0\end{equation}$$ The characteristic equation associated to this differential equation is $$\begin{equation}s^2 + 2\alpha s + \omega_0^2 = 0\end{equation}$$ whose roots are given by $$\begin{equation}s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}\end{equation}$$ Depending on the values of $\alpha$ and $\omega_0$, the discriminant can be positive, negative or zero and we distinguish three different cases:

• If $\alpha > \omega_0$, the roots $s_1$ and $s_2$ are real and distinct. In this case, the natural response is overdamped and is given by $$\begin{equation}y_n(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}\end{equation}$$
• If $\alpha = \omega_0$, the roots $s_1$ and $s_2$ are real and equal. In this case, the natural response is critically damped and is given by $$\begin{equation}y_n(t) = (A_1 + A_2 t) e^{-\alpha t}\end{equation}$$
• If $\alpha \lt \omega_0$, the roots $s_1$ and $s_2$ are complex conjugates with the real part equal to $\alpha$ and the imaginary part equal to $\sqrt{\omega_0^2 - \alpha^2}$. In this case, the natural response is underdamped and is given by $$\begin{equation}y_n(t) = e^{-\alpha t} \left[ A_1 \cos(\omega_d t) + A_2 \sin(\omega_d t) \right]\end{equation}$$ where $\omega_d = \sqrt{\omega_0^2 - \alpha^2}$ has the dimensions of a frequency and is called the damped natural frequency of the circuit.

When the forcing functoin $f(t)$ in the second-order differential equation is a constant (say $f(t) = F_0$), the particular solution is also a constant given by $$\begin{equation}y_p(t) = Y_0\end{equation}$$ where $Y_0=\frac{F_0}{\omega_0^2}$. Therefore, the complete response of the circuit in this case is $$\begin{equation}y(t) = y_n(t) + \frac{F_0}{\omega_0^2}\end{equation}$$ where $y_n(t)$ is the natural response discussed in the previous section. Notice that when $t$ goes to infinity (i.e., after a very long time), the natural response vanishes and the solution approaches the constant value $Y_0$.

When the forcing function $f(t)$ is not equal to a constant, the particular solution $y_p(t)$ can be found using various methods such as the method of undetermined coefficients or variation of parameters. The complete response of the circuit is still given by $$\begin{equation}y(t) = y_n(t) + y_p(t)\end{equation}$$ where $y_n(t)$ is the natural response.

In general, it is relatively difficult if not impossible to find the particular solution for an arbitrary forcing function analytically. However, if the right-hand side of the differential equation is a linear combination of functions whose derivatives are similar to the original functions (such as polynomials, exponentials, sines, and cosines), one can use the method of undetermined coefficients to find the particular solution.

In general, to solve a second-order transient problem in an RLC circuit, we need to write the differential equation, find the natural response, find the particular response, combine the responses, i.e., $y(t) = y_n(t) + y_p(t)$, and use the initial conditions to solve for the unknown constants in the natural response.

However, since we already know form of the currents and voltages in the circuit we can directly use these expression to find the complete solution without going through all the steps. Instead, we proceed as follows:

1. Compute the values of the damping factor $\alpha$ and the undamped natural frequency $\omega_0$ from the circuit parameters. Notice that, depending on whether the configuration of the circuit is series or parallel we have
   • Parallel RLC circuit: $$\alpha = \frac{1}{2RC}, \quad \omega_0 = \frac{1}{\sqrt{LC}}$$
   • Seris RLC circuit: $$\alpha = \frac{R}{2L}, \quad \omega_0 = \frac{1}{\sqrt{LC}}$$
2. Write the general form of the solution (either voltage or current) based on whether the forcing function is constant or not. If the forcing function is constant or zero (which is often the case in circuit problems) we have:
   • If $\alpha > \omega_0$, the roots $s_1$ and $s_2$ are real and distinct. In this case, the natural response is overdamped and is given by $$\begin{equation}y(t) = Y_0 + A_1 e^{s_1 t} + A_2 e^{s_2 t}\end{equation}$$ where $s_1$ and $s_2$ are the roots of the characteristic equation $s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}$.
   • If $\alpha = \omega_0$, the roots $s_1$ and $s_2$ are real and equal. In this case, the natural response is critically damped and is given by $$\begin{equation}y(t) = Y_0 + (A_1 + A_2 t) e^{-\alpha t}\end{equation}$$
   • If $\alpha \lt \omega_0$, the roots $s_1$ and $s_2$ are complex conjugates. In this case, the natural response is underdamped and is given by $$\begin{equation}y(t) = Y_0 + e^{-\alpha t} \left[ A_1 \cos(\omega_d t) + A_2 \sin(\omega_d t) \right]\end{equation}$$ where $\omega_d=\sqrt{\omega_0^2-\alpha^2}$.
   If the forcing function is not not constant, then $Y_0$ in the above equations becomes a function of time, which needs to be determined using (for instance) the method of undetermined coefficients.
3. Compute constant $Y_0$ using the solution of the circuit at $t=\infty$ (if the forcing function is constant or zero). Notice that the natural response vanishes at $t=\infty$ because the exponential goes to $0$; therefore $y(\infty)=Y_0$.
4. Apply the initial conditions to find constants $A_1$ and $A_2$. These initial conditions can be found by analyzing the circuit at $t=0^-$ and imposing that the voltage across capacitors and the current through inductors cannot change instantaneously.