Nodal Analysis with Voltage Sources

Nodal analysis is a remarkably effective method used in electrical engineering to simplify the analysis of complex circuits, including those with dependent or independent voltage sources. Its strength lies in its systematic approach to breaking down circuits into manageable components, making it easier for engineers to understand and solve.

Consider a circuit that contains four resistors and two voltage sources, as shown in Figure 1. One of these voltage sources is connected between a non-reference node and the reference node. In this configuration, the voltage at the non-reference node can be written directly as equal to the voltage of the source. This simplifies the problem by reducing the number of unknowns.

The other voltage source is connected between two non-reference nodes. This configuration forms what is known as a supernode or generalized node. Supernodes are a unique concept in nodal analysis, which helps to handle situations where voltage sources are connected between non-reference nodes.

Analyzing a circuit with a supernode necessitates the application of both Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL). First, KCL is applied to the supernode. This law states that the sum of currents entering a node equals the sum of currents leaving it. By considering the currents through each element within the supernode, an equation can be obtained and written in terms of the node voltages.

Next, to apply KVL, the circuit is redrawn to highlight the loop containing the supernode. KVL states that the sum of the electromotive forces in any closed loop or mesh in a network is equal to the sum of the potential drops in that loop. Going around this loop in a clockwise direction gives a constraint equation.

With these steps, three equations are obtained, which represent the behavior of the circuit. These equations can be simultaneously solved to determine the node voltages.