The currents and voltages in electric circuits can be found by solving a system of equations obtained by writting Kirchhoff’s voltage and current laws (e.g. nodal or mesh analysis). If the circuit contains only voltage and current sources (whose values can change in time) and resistors, inductors and capacitors with fixed values (i.e. the values of R, L, and C do not change in time), this system of equations becomes a system of linear integro-differential equations with constant coefficients. It turns out that the solution of such a system is the superpositon of the homogeneous and prticular solutions. The homogeneous solution gives the natural response of the circuit (also called transient response), while the particular solution gives the forced response of the system. Since the voltage and current sources appear as free terms in the mesh and nodal analysis equations, the homogeneous solution does not depend on the waveforms of the voltage and current sources. Therefore, the natural response of any linear circuit does not depend on the voltage and currents sources and can be computed by seeting them to zero.

Notice that, when we analyzed AC circuits using the formalism of complex analysis, we were assuming that the circuit was driven by the current and voltage sources for long period of time and the transient response "died out". In this way, we were actually computing only the forced response of the system, which was sinusoidal.

In this chapter, we are interested in computing the natural and transient responses of electric circuits. For this purpose, we will first need to learn how to write the system of integro-differential equations and, then, how to solve it. Although systems of integro-differential equations are slightly harder to solve than systems of algebraic equations, since the equations are linear, the system can be reduced to a system of real equations (using the formalism of Laplace transform).

Most methods that are used to study DC and AC circuits can also be used to analyze time-dependent circuits: Ohm's law, current division, voltage division, nodal analysis, mesh analysis, and superposition. It is also worth noticing that, in the case of DC circuits these methods lead to systems of equations with real coefficients; in the case of AC circuits they lead to systems of equations with complex coefficients; in the case of transient circuits, they lead to systems of system of integro-differential equations.

In general, a transient circuit is a circuit that is initially at steady-state and, when perturbed (say at $t=0$), it goes through some transient regime in which the potentials and currents in the circuit will change, after which it reaches another steady-state. The circuit will remain in this final steady-state until it is perturbed again. Notice that, in order for a circuit to reach a final steady-state, the current and voltage sources in the circuit needs to be time-independent (at least after a certain period of time); otherwise, the potentials and currents in the circuit will vary in time.

Depending on the order of the differential equation that describes the nodal potentials and branch currents in the circuit, transient circuits can be of first-order, second-order, or higher-order. In general, if the circuit contains $n$ storage elements (i.e. inductors and capacitors), the circuit will be at most of the $n$-th order.

Since first-order transient circuits are described by a first-order differential equation (usually with constant coefficients). For this reason, the solution of the first-order circuits can be often computed analytically (see First-Order Transients).

Ohm's law holds for any resistor at any moment in time. Using the notations in Fig. 1, we have $$\begin{equation}V(t) = R~I(t)\end{equation}$$ or $$\begin{equation}I(t)=\frac{V(t)}{R}\end{equation}$$

Power is a quantity defined at any moment in time. In the case of time-dependent circuits, it is recommended to call it instanteneous power (instead of just power) to distinguish it from average power, reactive power, and complex power. As we have discussed in the first chapter, the instanteneous power dissipated by any 2-terminal component is

$$\begin{equation}P_d(t)=V(t) I(t)\end{equation}$$

where voltage $V(t)$ and current $I(t)$ are shown in Fig. 3. The power generated by a 2-terminal component is

$$\begin{equation}P_g(t)=-P_d(t)=-V(t) I(t)\end{equation}$$