The currents and voltages in electric circuits can be found by solving a system of equations obtained by writing 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. The solution of such a system is the superposition of the homogeneous and particular solutions. The homogeneous solution gives the natural response of the circuit (also called the transient response), while the particular solution gives the forced response of the system. Because the voltage and current sources appear as free terms in the mesh and nodal analysis equations, the homogeneous solution does not depend on the waveform of those sources. Therefore, the natural response of any linear circuit can be computed by setting the independent sources to zero.

Notice that, when we analyzed AC circuits using phasor methods, we implicitly assumed that the circuit had been driven by the sources for a long time and the transient response had died out. In that case we were computing only the forced (steady-state) sinusoidal response of the system.

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. An important way of solving these equations will be the method of Laplace transforms.

Most methods that are used to study DC and AC circuits (i.e. Ohm's law, current division, voltage division, nodal analysis, mesh analysis, and superposition) can also be used to analyze time-dependent circuits. It is also worth noting that, for DC circuits these methods lead to systems of equations with real coefficients; for AC circuits they lead to systems with complex coefficients; and for transient circuits they lead to 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 change, eventually reaching another steady-state. The circuit will remain in this 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 need to be time-independent (at least after a certain period of time); otherwise, the potentials and currents in the circuit will keep varying 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.

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).

Using the notations in Fig. 1, Ohm's law can be written as $$\begin{equation}V(t) = R~I(t)\end{equation}$$ or $$\begin{equation}I(t)=\frac{V(t)}{R}\end{equation}$$

Power is a time-dependent quantity. In the case of time-dependent circuits, it is recommended to call it instantaneous 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 instantaneous power dissipated by any 2-terminal component is

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