Nodal Analysis

Nodal analysis is a fundamental method in electrical engineering used to simplify the process of circuit analysis. This method revolves around the concept of using node voltages as the primary variables for circuit analysis. The objective is to determine the voltage at each node in a circuit, which can then be used to find other quantities of interest, such as currents through specific components.

Consider, for instance, a simple circuit composed of three nodes and three resistors, as shown in Figure 1. The first step in nodal analysis involves selecting a reference or datum node. This node is typically chosen based on convenience, and its voltage is assigned a value of zero.

Figure 1

The subsequent nodes, referred to as non-reference nodes, are assigned nodal voltages relative to this reference node. In the example considered here, there are two non-reference nodes labeled 1 and 2, each with their respective node voltages.

To establish a relation between the branch currents, Kirchhoff's Current Law (KCL) is applied to the non-reference nodes. KCL states that the algebraic sum of currents entering a node (or a closed boundary) is zero. This law is grounded on the principle of charge conservation – that is, a charge cannot be created or destroyed.

Following the application of KCL, Ohm's Law is used to express the branch currents passing through the three resistors in terms of the node voltages. Ohm's Law postulates that the current passing through a conductor between two points is directly proportional to the voltage across the two points.

With the branch currents expressed in terms of node voltages, these values are substituted into the equations derived from KCL. This substitution results in two simultaneous equations since, for a circuit with 'n' nodes, 'n-1' independent equations are obtained.

Lastly, if the values of the resistors and source currents are known, they can be substituted into the two equations. Solving these equations will yield the node voltages. This information is invaluable, as it can help in understanding the behavior of the circuit and in designing or troubleshooting electrical circuits.

In conclusion, nodal analysis is a powerful tool in circuit analysis, providing a systematic method to determine the distribution of voltages within a circuit, which can then be used to calculate other parameters like currents and power.