Instead of solving the system of nodal or mesh equations in the time domain (which leads to integro-differential equations), it is often easier to transform the original circuit into the s-domain. Then we solve the nodal or mesh analysis equations in the s-domain and, finally, take the inverse Laplace transform to obtain the time-domain variables.

Note that the system of equations for the transformed circuit (in the s-domain) is equivalent to the system obtained by writing the time-domain integro-differential equations and then taking their Laplace transform. Learning the rules to convert circuit elements to the s-domain therefore skips the intermediate step of forming and transforming the integro-differential equations.

When we write KCL in a nodal analysis, we express branch currents in terms of nodal potentials. Table 1 shows how to write the currents through various components in the s-domain. In this table, $u(t)$ denotes the unit step function

$$\begin{equation}u(t)=\begin{cases}0, & \text{if }t<0\\1, & \text{if }t\ge 0\end{cases}\end{equation}$$

Table 1. Currents going through various components in terms of nodal potentials in the s-domain. The voltage across each component is $V(s)=v_{+}(s)-v_{-}(s)$, where $v_{+}$ and $v_{-}$ are the nodal potentials of the $+$ and $-$ in the s-domain. In the last two equations, $V_{C}(0)$ is the voltage across the capacitor and $I_L(0)$ is the current going through the inductor at $t=0$ (in time-domain). These initial values need to be computed in advance, before starting the analysis in the s-domain.

Name	Symbol	Branch current (s-domain)
Independent current source		$$\begin{equation}I(s)=\frac{I_0}{s}\end{equation}$$
Resistor		$$\begin{equation}I(s)=\frac{V(s)}{R}\end{equation}$$
Capacitor		$$\begin{equation}I(s)=sC\cdot V(s)-C\cdot V_{C}(0)\end{equation}$$
Inductor		$$\begin{equation}I(s)=\frac{V(s)}{sL} + \frac{I_{L}(0)}{s}\end{equation}$$

For instance, the nodal equation in the s-domain for node $v_{3}$ for the circuit represented in Fig. 1 is

$$\begin{equation}sC\cdot (v_{3}-v_{2}) - C\cdot V_{C0} + \frac{I_{L0}}{s} + \frac{v_3-v_5}{sL_1} + \frac{v_{3}-v_{2}}{R_{1}}=0\end{equation}$$

where $I_{L0}$ is the current through the inductor at $t=0$. The previous equation is valid for $t>0$. To simplify notation, we have not denoted explicitly that $v_1$,..., $v_5$ are functions of $s$; they should be understood as $v_1(s)$,..., $v_5(s)$.

When we write KCL in a nodal analysis, we need to express the branch currents in terms of the nodal potentials. Table 2 describes how to write the currents going through different components in nodal analysis.

Table 2. Voltages across various components as a function of the current going through the component in the s-domain. In the last two equations, $V_{C}(0)$ is the voltage across the capacitor and $I_L(0)$ is the current going through the inductor at $t=0$ (in time-domain). These initial values need to be computed in advance, before starting the analysis in the s-domain.

Name	Symbol	Voltage
Independent voltage source		$$\begin{equation}V(s)=\frac{V_{0}}{s}\end{equation}$$
Resistor		$$\begin{equation}V(s)=I(s){R}\end{equation}$$
Capacitor		$$\begin{equation}V(s)=\frac{I(s)}{sC} + \frac{V(t_0)}{s}\end{equation}$$
Inductor		$$\begin{equation}V(s)=sL\cdot I(s)-L\cdot I_{L}(0)\end{equation}$$

The mesh equation in the s-domain for mesh current $i_{2}$ in the circuit represented in Fig. 2 is

$$\begin{equation}i_{2}R_{2}+\frac{V_{C0}}{s}+\frac{i_2-i_3}{sC_1} -V_1 -L\cdot I_{L0} + sL_1\cdot (i_{2}-i_{1})=0\end{equation}$$

where $V_{C0}$ is the capacitor voltage at $t=0$. The previous equation is valid for $t>0$, and $i_1$, $i_2$, and $i_3$ should be understood as functions of $s$.

The algorithm for writing the nodal or mesh analysis equations in the s-domain is the same as in the DC, AC, or time-domain cases. See the linked examples to learn how to write and solve the nodal and mesh equations in the s-domain.