Nodal & Mesh Analysis | Circuit Analysis for GATE ECE

Basic Circuit Terminology

Before studying Nodal and Mesh Analysis, it is important to understand the basic terminology used to describe an electrical circuit.

Node

A node is a point or region in a circuit where two or more circuit elements are connected and therefore have the same electrical potential.

Branch

A branch is a portion of a circuit containing one or more elements connected between two nodes. The same branch current flows through all series elements within that branch.

Mesh

A mesh is a closed path in a planar circuit that does not contain any other closed path within it. Meshes are used as the basis of Mesh Analysis.

Independent Meshes

For a connected planar circuit, the number of independent meshes is related to the number of branches and nodes by:

$M=B-(N+1)$

where $M$ is the number of independent meshes, $B$ is the number of branches, and $N$ is the number of nodes.

Nodal Analysis: Node Voltage Method

Nodal Analysis, also known as the Node Voltage Method, is a systematic technique used to determine unknown node voltages in an electrical circuit.

The method is based on Kirchhoff's Current Law (KCL). The branch currents are expressed in terms of node voltages using Ohm's Law.

$\sum I=0$

For a resistor $R$ connected between two nodes having voltages $V_1$ and $V_2$, the branch current can be written as:

$I=\frac{V_1-V_2}{R}$

Thus, the basic steps of Nodal Analysis involve:

• Selecting a reference node and assigning it $0\,V$.
• Assigning voltage variables to the remaining non-reference nodes.
• Applying KCL at the required nodes.
• Using Ohm's Law to express branch currents in terms of node voltages.
• Solving the resulting simultaneous equations.

Number of Nodal Equations

If a circuit has $N$ nodes and one node is selected as the reference node, the number of independent nodal equations is generally:

$N-1$

Supernode

A supernode is formed when a voltage source is connected between two non-reference nodes.

In a supernode, KCL is applied around the outer boundary of the supernode. The voltage source provides an additional voltage constraint between the two node voltages.

Therefore, Supernode Analysis involves:

• KCL for the supernode.
• Ohm's Law for resistive branches connected to the supernode.
• A voltage constraint provided by the voltage source.

For example, if a voltage source $V_s$ is connected between nodes having voltages $V_1$ and $V_2$, the constraint may be written as:

$V_1-V_2=V_s$

Mesh Analysis: Mesh Current Method

Mesh Analysis, also known as the Mesh Current Method, is a systematic technique used to determine unknown mesh currents in a planar circuit.

The method is based on Kirchhoff's Voltage Law (KVL). The voltage drops across resistive elements are expressed using Ohm's Law.

$\sum V=0$

For a simple mesh containing a voltage source $V$ and resistors $R_1$ and $R_2$, if the assumed mesh current is $I_1$, applying KVL gives:

$V-I_1R_1-I_1R_2=0$

Thus, the basic steps of Mesh Analysis involve:

• Assigning a mesh current to each independent mesh.
• Applying KVL around each independent mesh.
• Using Ohm's Law to express resistor voltage drops in terms of mesh currents.
• Solving the resulting simultaneous equations.

Number of Mesh Equations

For a connected planar circuit, if there are $B$ branches and $N$ nodes, the number of independent meshes is:

$M=B-(N+1)$

Therefore, Mesh Analysis generally requires $M$ independent mesh equations.

Shared Elements in Mesh Analysis

When two adjacent meshes share a common resistor, the current through that resistor is determined by the difference between the corresponding mesh currents.

$I_R=I_1-I_2$

Using Ohm's Law, the voltage drop across the shared resistor is:

$V_R=R(I_1-I_2)$

The sign depends on the assumed directions of the mesh currents and the direction in which KVL is written.

Supermesh

A supermesh is formed when a current source lies in the common branch between two adjacent meshes.

Since the voltage across an ideal current source is not directly known, KVL is applied around the outer boundary of the supermesh, excluding the current-source branch.

The voltage drops across the resistive elements are determined using Ohm's Law, while the current source provides an additional constraint relating the mesh currents.

Therefore, Supermesh Analysis involves:

• KVL around the outer boundary of the supermesh.
• Ohm's Law for the resistive elements.
• A current constraint provided by the current source.

For example, depending on the assumed mesh-current directions, the current-source constraint may be:

$I_1-I_2=I_s$

Key Points for GATE

• Node: A point or region where two or more circuit elements are connected and have the same electrical potential.
• Branch: A portion of a circuit connected between two nodes.
• Mesh: A closed path in a planar circuit that contains no other closed path within it.
• Nodal Analysis: Uses KCL + Ohm's Law to determine unknown node voltages.
• Supernode: Uses KCL + Ohm's Law + voltage-source constraint.
• Mesh Analysis: Uses KVL + Ohm's Law to determine unknown mesh currents.
• Supermesh: Uses KVL + Ohm's Law + current-source constraint.
• Reference node: In Nodal Analysis, one node is selected as the reference and assigned $0\,V$.
• Mesh Analysis limitation: Mesh Analysis is directly applicable to planar circuits.
• Nodal Analysis: Can be applied to both planar and non-planar circuits.
• Number of independent nodal equations: For $N$ nodes, one reference node leaves generally $N-1$ independent nodal equations.
• Number of independent mesh equations: For a connected planar circuit with $B$ branches and $N$ nodes, $M=B-N+1$ independent meshes exist, so $M$ independent mesh equations are generally required.

Where Do Nodal Analysis and Mesh Analysis Apply?

Nodal Analysis and Mesh Analysis have different requirements when applied to electrical circuits. The following table summarizes their applicability under common GATE ECE circuit conditions.