Nodal Analysis in Electric Circuits (Example With Step by Step Analysis)

What Is Nodal Analysis?

Nodal analysis, also called the node-voltage method, solves a circuit in terms of the voltages at its nodes. One node is assigned zero volts, and every other node voltage is measured relative to it.
The main features of nodal analysis are:

• Nodal analysis is based on the Kirchhoff’s Current Law (KCL).
• A circuit with n nodes normally has n – 1 unknown node voltages after one node is selected as the reference.
• Solving the simultaneous node equations gives those unknown node voltages.
• Each unknown node voltage needs an independent equation, although voltage-source constraints can replace some individual KCL equations.

Reference and Non-Reference Nodes

• A non-reference node has an unknown or specified voltage measured relative to the reference node. Node 1 and Node 2 in the example below are non-reference nodes.
• Reference node – This node is assigned zero volts and provides the common voltage reference for the circuit. Other names are datum node and ground node.

Reference-Node Symbols

1. Chassis ground – This symbol identifies a common connection to equipment chassis or frame.
   
2. Earth ground – This symbol identifies a node intentionally connected to the earth.

How to Solve a Circuit with Nodal Analysis

Basic Steps in Nodal Analysis

1. Choose one node as the reference and label the other node voltages V₁, V₂… V_n-1, all measured relative to that reference.
2. Choose one current sign convention, then apply KCL at each node whose voltage is unknown.
3. Use Ohm’s law to express each resistor current in terms of the voltages at its two ends.

For a resistor between two nodes, define the branch direction first. The branch current in that direction equals the voltage at the starting node minus the voltage at the ending node, divided by the resistance.

IV. Substitute these current expressions into the KCL equations so that the unknowns are node voltages and the coefficients depend on the resistances.

V. Solve the simultaneous equations for the unknown node voltages. Check the result by substituting the values back into the KCL equations.

Nodal Analysis with Current Sources

Nodal analysis with current sources is direct because a source current enters the KCL equation with its known value and chosen sign.

Example: Calculate the node voltages in the following circuit.

The circuit has three nodes. One is the reference node, and the other two are the non-reference nodes labelled Node 1 and Node 2.

Step I. Label the unknown node voltages v₁ and v₂. Choose and mark a direction for each branch current.

Step II. Apply KCL at Node 1 and Node 2.
KCL at Node 1:
KCL at Node 2:
Step III. Apply Ohm’s law to the resistor currents in each KCL equation.
• Apply Ohm’s law to the KCL equation at Node 1.

Simplifying gives:

• Apply Ohm’s law to the KCL equation at Node 2.

Simplifying gives:

Step IV. Solve Equations 3 and 4 simultaneously for v₁ and v₂.
Using elimination:

Substituting v₂ = 20 V into Equation 3 gives:

The node voltages are v₁ = 13.33 V and v₂ = 20 V.

Nodal Analysis with Voltage Sources

Case I. If a voltage source is connected between the reference node and a non-reference node, that source directly sets the node voltage, with its sign determined by source polarity. Apply KCL to the remaining unknown-voltage nodes as in the current sources case. In this example, v₁ = 10 V.

Case II. If a voltage source connects two non-reference nodes, treat the source and those nodes as a supernode.

Supernode Analysis

What Is a Supernode?

A supernode is formed when an independent or dependent voltage source connects two non-reference nodes. The supernode boundary encloses the source and both nodes.

In this figure, the 5 V source connects Node 2 and Node 3, so those nodes and the source form one supernode for the KCL equation.

Properties of a Supernode

• The voltage source gives a known relationship between the two node voltages, including its polarity.
• A supernode is an analysis boundary, not an extra node-voltage unknown.
• Solving a supernode requires KCL around its boundary and a voltage constraint from KVL.
• An element can be connected in parallel with the voltage source and will share the same terminal voltage.
• KCL applies to the whole supernode boundary just as it applies to an ordinary node.

How to Solve a Circuit Containing a Supernode

This example shows how to solve a circuit containing a supernode.

The 2 V source connects Node 1 and Node 2 and forms a supernode. A 10 Ω resistor is connected in parallel with the source.
Note – The ideal voltage source fixes v₂ – v₁ = 2 V, so the parallel resistor does not change that voltage constraint. Its current is internal to the supernode and therefore cancels from the KCL equation around the supernode boundary, although the resistor can still carry current. Applying KCL around the boundary gives:

Express the branch currents in terms of the node voltages.



Combine Equation 5 with the 2 V source constraint in Equation 6.

The solution is v₁ = -7.333 V and v₂ = -5.333 V. Their difference is 2 V, as required by the source polarity.