Short Answer:

Mesh analysis is a method used in electrical circuit analysis to calculate the current flowing through different loops (meshes) of a circuit. It is based on Kirchhoff’s Voltage Law (KVL), which states that the sum of voltages around any closed loop in a circuit is zero. This method is mainly used in planar circuits where no branches cross over each other.

In mesh analysis, we assign a current to each mesh and write KVL equations to form a system of equations. Solving these equations helps find the unknown mesh currents, making it easier to analyze complex circuits with multiple loops.

Detailed Explanation:

Mesh analysis

Mesh analysis, also known as the loop current method, is a systematic technique in circuit theory used to find unknown currents in a planar circuit. A mesh is a loop in a circuit that does not enclose other loops. The method simplifies the process of analyzing multiple loops by focusing only on mesh currents instead of individual branch currents.

The technique works by applying Kirchhoff’s Voltage Law (KVL) around each mesh. According to KVL, the algebraic sum of all voltages in a closed loop is zero. By applying this law to each mesh and solving the resulting equations, we can determine the current in each mesh. From these mesh currents, other voltages and branch currents in the circuit can also be easily calculated.

Steps to perform mesh analysis

1. Identify the meshes
   • Look for independent loops in the circuit that do not contain smaller loops inside.
   • Label each mesh with a current variable (e.g., I1,I2,I3I_1, I_2, I_3I1​,I2​,I3​, etc.).
2. Assign mesh current direction
   • Usually, mesh currents are assumed to circulate in the clockwise direction, but any consistent direction is acceptable.
3. Apply KVL to each mesh
   • For each mesh, sum the voltage drops and rises across each component, keeping signs consistent.
   • Use Ohm’s Law (V=IRV = IRV=IR) for resistors.
4. Form equations
   • Each mesh gives one linear equation involving the mesh currents.
   • If two meshes share a resistor, the voltage drop across that resistor depends on the difference of mesh currents.
5. Solve the equations
   • Use substitution, elimination, or matrix methods to solve the system of equations and find the mesh currents.
6. Find branch currents and voltages (if needed)
   • Once mesh currents are known, use them to find other quantities in the circuit.

Example concept

Suppose a circuit has two meshes with resistors and voltage sources. Assign currents I1I_1I1​ and I2I_2I2​ to each mesh. If both meshes share a resistor RRR, the voltage across that resistor is R(I1−I2)R(I_1 – I_2)R(I1​−I2​) if both currents flow through the resistor in opposite directions. Apply KVL to each mesh and solve the resulting equations.

When mesh analysis is useful

• Best suited for planar circuits (circuits that can be drawn without crossing branches).
• Especially useful in resistive networks with multiple loops.
• Helps simplify complex networks into a small number of equations.

Advantages of mesh analysis

• Reduces the number of unknowns to mesh currents only.
• Simple to use for loop-based circuit analysis.
• Can be easily extended using matrix methods for larger networks.

Limitations of mesh analysis

• Not ideal for non-planar circuits (with overlapping branches).
• Becomes complex when circuits contain many voltage or current sources, especially dependent sources.
• Requires modification when current sources are shared between meshes (supermesh concept).

Conclusion:

Mesh analysis is a powerful method for solving circuits with multiple loops by applying Kirchhoff’s Voltage Law to each mesh. It simplifies the process by reducing the number of unknowns and focusing only on mesh currents. This method is especially useful for planar circuits and helps in systematically analyzing and solving electrical networks.