Fundamentals of Electricity: Mesh Analysis Study Notes

EE1301 Fundamentals of Electricity: Circuit Analyzing Techniques

Unit 2: Circuit Analyzing Techniques

2.2 Mesh Analysis
  • Definition of Mesh Analysis: Mesh analysis applies Kirchhoff's Voltage Law (KVL) to find unknown currents in electrical circuits.
  • Mesh: A mesh is defined as a loop or closed path in an electrical circuit that does not contain any other loops or enclosed paths within it.
  • Applicability: Mesh analysis is applicable solely to planar circuits.
    • Planar Circuit: A circuit that can be drawn on a two-dimensional plane without any of its branches crossing one another.
    • Non-Planar Circuit: A circuit that cannot be represented on a plane without overlaps or crossings of wires.

2.2.1 Mesh Analysis without Current Sources

Steps to Determine Mesh Current:
  1. Assign Mesh Currents: Designate mesh currents as $i1$, $i2$, …, $i_n$ for each of the $n$ meshes in the circuit.
  2. Apply KVL: Use Kirchhoff's Voltage Law to write equations for each mesh, focusing on the voltages across the elements in terms of the mesh currents.
  3. Use Ohm’s Law: Express the voltages in reference to the mesh currents according to Ohm’s law, where $V = IR$. This ensures that the voltages can be described in terms of the currents assigned to each mesh.
  4. Solve Equations: Solve the resultant set of $n$ simultaneous equations derived from the KVL application to determine the values of the mesh currents.
Example 2.4: Mesh Analysis Example
  • Objective: Find the branch currents $I1$, $I2$, and $I_3$ using mesh analysis.
  • Note: It is important to differentiate that mesh currents and branch currents are distinct in the context of circuit analysis.
    • Mesh Currents: Currents flowing in the defined meshes.
    • Branch Currents: Currents flowing in the physical branches of the circuit, which may be a function of the mesh currents.

2.2.2 Mesh Analysis with Current Sources

Case 1: Current Source in One Mesh
  • Situation: When a current source is present exclusively within a single mesh, the mesh current can be set equal to the source current.
    • Example: If the mesh current is denoted as $i2$, then $i2 = -5A$.
  • Mesh Equation: The mesh equation for any other mesh can be formulated in the standard approach, using KVL and the mesh currents defined.
Case 2: Current Source Between Two Meshes
  • Creation of Supermesh: When a current source appears between two meshes, a supermesh is created. This is achieved by excluding the current source and any series elements connected to it from the mesh consideration.
    • Supermesh Characteristics: A supermesh does not possess its own defined current, but it still abides by KVL. The voltages around the supermesh must sum up to zero just as they would in standard mesh analysis.
Example 2.5: Finding Currents Using Mesh Analysis
  • Objective: Calculate the mesh currents $i1$ to $i4$ using the methods of mesh analysis discussed previously.

Summary of Key Concepts

  • Mesh analysis is a systematic procedure that utilizes KVL to analyze circuits and calculate unknown currents.
  • Differentiation between mesh currents and branch currents is critical for accurate circuit representation and analysis.
  • The presence of current sources introduces specific adaptations to the standard mesh analysis techniques, including the implementation of supermeshes in cases where a current source connects multiple meshes.