Nodal Analysis: Nodal analysis is a method of network analysis particularly suited for networks having many parallel circuits with a common ground connected node. This method of network analysis is based on Kirchhoff’s first law or KCL. Nodal analysis significantly reduces the computational work required by reducing the set of equations to be solved in a particular circuit or network. Nodal analysis is also termed the node-voltage theorem.

Application of Nodal Analysis

Application of nodal analysis requires taking a node in the circuit as a reference or datum node. For simplicity and easy approach, the node has the highest number of circuit elements connected.

Each node I assigned with unknown voltage variables such as V1, V2, and so on. The voltage for the reference node is taken as 0 Volts.

Kirchhoff’s current law is applied at each of these nodes to form a system of linear algebraic equations. For a circuit consisting of ‘N’ number of nodes, by nodal analysis the number of equations thus formed will be N-1.

This system of linear equations is solved to determine the unknown nodal voltages.

Illustration of Steps Involved in Nodal Analysis

For the illustration of the analysis of the network using nodal analysis, we will consider an electrical circuit as shown below.

Nodal Analysis

Figure: Circuit for Illustration of Nodal Analysis

At first, we determine nodes in the given circuit and will take one of the nodes as a reference node. The node with the greatest number of circuit elements linked can be used as a reference node for ease of use. For the above circuit, we will take node C as a reference node as it is the junction for source EMF E1 and E2 along with resistor R4 and R5 making it the node with the highest number of connections of circuit elements.

Secondly, we will assign variables for the unknown node voltages. Such as VA and VB be the unknown node voltage for nodes A and B. While the voltage for reference node C is taken to be 0 Volts.

Thirdly, we will apply Kirchhoff’s Current Law for each node (excluding the reference node).

Such as in the above circuit.

Applying KCL for Node A which is the junction of resistors R1, R2, and R4.

Applying KCL for Node B which is the junction of resistor R2, R3, and R5.

Finally, solve the linear equation to find the unknown node voltages. Such as in the above case the two systems of linear equations (i) and (ii) can be solved to find the unknown node voltages VA and VB.

Examples

Example-1 Nodal Analysis

For the circuit shown below, we will determine the current flowing through 1 Ohm resistor using nodal analysis.

Nodal Analysis

Figure: 1.1

At first, we will take a node as a reference node. For our simplicity, we will take node C as a reference node as shown below.

Nodal Analysis

Figure: 1.2

Now, assigning the voltages at a node as VA and VB.

Then applying KCL at nodes A and B.

For Node A,

For Node B,

Now, solving equations (i) and (ii)

Current through 1 Ohm resistor is given by

The negative sign indicates the current is flowing from node B to A.

Example-2:

For the circuit shown below, we will determine the node voltages for nodes 1 and 2 using nodal analysis.

Nodal Analysis

Figure: 2.1

For our easiness, we will arbitrarily assume the current distribution as shown below.

Nodal Analysis

Figure: 2.2

Now, applying KCL at node 2 we have

So, we have


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