mesh analysis
A method of solving for the currents in an electric circuit, in which a system of equations is written for mesh currents (using Kirchhoff's voltage law) around each mesh (loop).
mesh analysis: systematic current solving via loop equations
Mesh analysis is a circuit analysis technique that uses Kirchhoff's voltage law to write equations for the currents flowing around each closed loop in a network. Instead of solving for currents through individual branches, you assign a current variable to each mesh (an independent loop) and write one equation per mesh. The sum of voltage drops around each loop must equal zero. This method converts a circuit into a system of simultaneous linear equations, which you then solve algebraically or using matrix methods.
The method works best on planar circuits, where all components lie in a single plane and no branches cross over each other. Start by identifying all independent meshes. In a circuit with b branches and n nodes, the number of independent meshes is b minus n plus one. For each mesh, traverse the loop in a chosen direction (usually clockwise), sum all voltage rises and drops using Ohm's law, and set the total to zero. When two meshes share a branch, the current through that branch is the difference between the two mesh currents.
Practical application and limits
Mesh analysis shines in circuits with series-parallel combinations and resistive networks. Engineers prefer it over nodal analysis when there are many voltage sources or when the circuit structure naturally suggests independent loops. However, the method becomes cumbersome with many meshes because the number of equations grows quickly. A circuit with six independent meshes requires solving six equations simultaneously, which is tedious without a computer.
The method struggles with nonplanar circuits or those containing current sources alone. Current sources do not fit easily into mesh equations; you must either convert them to Thevenin equivalent form or use modified mesh analysis. Similarly, controlled (dependent) sources complicate the equations because mesh currents must be expressed in terms of the control variables.
Mesh analysis forms one pillar of circuit theory alongside nodal analysis and network theorems like superposition and Thevenin's theorem. For linear DC or sinusoidal AC circuits, the two methods yield identical results but often require different effort depending on circuit topology. Modern circuit simulators use mesh analysis internally for certain problem classes, though they also employ other algorithms for speed and numerical stability.