The nodal analysis method that we introduced for DC and AC circuits can be generalized to time-dependent circuits. In this case, the method results into a system of integro-differential equations, which can be reduced to a system of ordinary differential equations (ODE) that needs to be solved to compute the time-dependent nodal potentials $v_i(t)$, where $i=1,...,n$. The system of nodal analysis equations contains:

• Voltage constrained equations (one equation for each voltage source)
• Kirchhoff's current law equations (one equation per node or supernode)

When we solve the system of nodal analysis equations in time-domain, we should always impose initial conditions for the current going throught the inductors and voltage across the capacitors (otherwise, this system would not have unique solution). These initial conditions are usually computed in advance.

The voltage constrained equations are written in the same way as in the case of DC circuits, the only difference being that the voltages of the sources are time-dependent: $$\begin{equation}V_i(t)=v_{i+}(t)-v_{i-}(t)\end{equation}$$ where $i=1,...,m$ and $v_{i+}(t)-v_{i-}(t)$ is the difference between the potentials of the positive and the negative electrodes of source $m$.

When we write KCL, the branch currents depend on the type of the device, as described in Table 1.

Table 1. Currents going through various components in terms of nodal potentials. The voltage across each component is $V(t)=v_{+}(t)-v_{-}(t)$.

Name	Symbol	Branch current (t-domain)
Independent current source		$$\begin{equation}I(t)\end{equation}$$
Resistor		$$\begin{equation}I(t)=\frac{V(t)}{R}\end{equation}$$
Capacitor		$$\begin{equation}I(t)=C\dfrac{dV(t)}{dt}\end{equation}$$
Inductor		$$\begin{equation}I(t)=I(t_0)+\frac{1}{L}\int_{t_0}^{t}V(t){dt}\end{equation}$$

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

$$\begin{equation}C\dfrac{d(v_{3}-v_{2})(t)}{dt} + I_{L0}+\frac{1}{L_1}\int_{0}^{t}(v_{3}-v_{5}){dt} + \frac{v_{3}-v_{2}}{R_{1}}=0\end{equation}$$

where $I_{L0}$ is the current going through the inductor at $t=0$. Notice that the previous equation is valid for $t>0$. To simplify notations, we have not denoted explicitly that $v_1$,..., $v_5$ are functions of $t$, however, we should consider them as $v_1(t)$,..., $v_5(t)$.

Assume we have a circuit with $n$ nodes (excluding the ground node), $m$ voltage sources and $c$ control variables.

Step 1. Identify the nodes in the circuit and select one of them as a reference node. The reference node is treated as ground node.

Step 2. Label the potentials at each of the $n$ nodes with $v_1$, $v_2$, ..., $v_n$, where $n$ is the number of nodes (excluding the ground node).

Step 3. Write the system of nodal analysis (ordinary differential) equations, which will have $n+c$ equations ($m$ voltage constrained equations, $n-m$ KCL equations, and $c$ equations for the control variables). It is a good practice two write the $n+c$ equations in the order specified below:

Step 4. Solve the system on nodal analysis equations to compute the $n$ nodal potentials and $c$ control variables, which, in general, will all now be time-dependent.

Step 5. Compute the sought variables.