Nodal analysis is a method based on Kirchhoff's current law that can be used to compute the potentials in a circuit. The technique is based on the fact that, if a circuit has $n+1$ nodes (including the ground node), the $n$ potentials (excluding the potential of the ground node, which is assumed to be $0$) can be found by solving a system of $n$ independent equations that can be obtained by applying KCL, in which the current going through each branch is computed based on the current-voltage characteristic of the element on that branch. If the circuit contains $m$ voltage sources (notice that in the case of voltage sources, the current vs. voltage is a multivalued function), $m$ equations will be the voltage constrained equations of each voltage source as described below.

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 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 to 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.

Step 5. Compute the sought variables.

• A supernode is formed by enclosing (or combining) two or more nodes that are connected by one or more voltage sources; the combined nodes should not include the ground node. Usually, we will need to write equations for supernodes whenever we have voltage sources that are not connected to the ground node directly of through any other voltage sources. Since a supernode contains voltage sources, a supernode does not have a single potential (in fact, it will contain multiple regular nodes).

  Fig. 1 shows an example of a circuit with one supernode formed by combining nodes $v_4$ and $v_5$. Note that the dependent voltage does not form a supernode because it is connected to ground through another voltage source ($V_1$). If $V_1$ was a resistor, the dependent source had formed another supernode. The KCL equation for the supernode shown in Fig. 1 is (notice that the crossed terms cancel each other) $$\begin{equation}\frac{v_4}{R_5}+\frac{v_5-v_3}{R_4}-I_1=0\end{equation}$$

  Fig. 2 shows an example of a circuit with one supernode formed by combining nodes $v_1$, $v_2$ and $v_4$. The KCL equation for the supernode shown in Fig. 2 is $$\begin{equation}\cancel{\frac{v_2-v_1}{R_1}}+\cancel{\frac{v_1-v_2}{R_1}}+\frac{v_1-v_3}{R_2}+\frac{v_2-v_3}{R_3}-\frac{v_5}{R_5}+=0\end{equation}$$

• The number of KCL equations that we write at steps 3.C and 3.D should be $n-m$.
• Sometimes, it is convenient to add the equations for the sought variables to the $n+c$ nodal analysis equations. In this case, the solution of the system will include contain the values of the sought variables.
• Nodal analysis can be used to analyze both linear and nonlinear circuits, DC and AC circuits, as well as time-dependent circuits. In the case of DC circuits we obtain a system of equations with real coefficients; in the case of AC circuits we obtain a system of equations with complex coefficients; in the case or time-dependent problems we obtain system of integro-differential equations (which can usually be reduced to a system of ordinary differential equations).

The sought variables can usually be expressed directly in terms of the nodal potentials.

• The voltage between two nodes can be expressed directly as the difference of the nodal potentials of the two nodes. For instance, the voltage $V_x$ in Fig. 1 is $V_x=0-v_4=-v_4$.
• The current going through a resistor can be computed using Ohm's law as a function of the voltage across the resistor and the resistance. For instance, the current going downwards through resistor $R_4$ in Fig. 2 is $(v_3-v_5)/R_4$.
• The current going out or in through a terminal of a voltage source can be computed indirectly, by writing KCL for the node which contains that terminal. In this way, we compute the sum of all currents entering and leaving the node, and by isolating the term associated with the voltage source we obtain the current flowing through its terminal. For instance, to compute current $I_0$ going through voltage source $V_2$ in Fig. 2, we can write that $$\begin{equation}I_0 = \frac{v_5-v_4}{R_4}-I_1\end{equation}$$

• The power dissipated by a resistor can be found by computing the voltage across the resistor, say $V_R$, and using $P=V_R^2/R$. For instance, the power dissipated by resistor $R_1$ in Fig. 2 is $(v_1-v_2)^2/R_1$.

• The power generated by a current source can be computed by multiplying the the current of the source by the voltage from node where the current goes out of the source to the node where current gets in the source. For instance, the power generated by source $I_1$ in Fig. 2 is $P_g=I_1(v_5-v_3)$

• The power generated by a voltage source can be computed by multiplying the the voltage of the source by the current going from the negative to the positive terminal of the source. For instance, the power generated by source $V_2$ in Fig. 2 is $P_g=V_2\cdot(-I_0)$ (where $I_0$ was computed above).