mesh

In circuit analysis, a mesh is a closed loop within an electrical network that contains no other loops inside it. This concept comes directly from how the circuit is drawn: imagine the circuit diagram laid out on a flat surface, and a mesh is any enclosed path that forms the perimeter of a region with no branches cutting through it. The mesh represents a route around which Kirchhoff's voltage law applies, meaning the algebraic sum of voltages around that loop must equal zero.

Mesh analysis is a systematic method for solving circuits with multiple loops. Rather than writing equations for every node and branch, you assign a mesh current to each independent loop, then write one voltage equation per mesh. For a circuit with three independent loops, you get three simultaneous equations relating the three mesh currents. This reduces complexity significantly compared to nodal analysis when the circuit has fewer loops than nodes.

A mesh differs from any closed path in the circuit. A closed path can pass through inner regions; a mesh cannot. In a planar circuit (one that can be drawn without crossing wires), the number of independent meshes equals the number of branches minus the number of nodes plus one. This relationship is fixed by topology, not by component values. A circuit with six components forming two side-by-side loops has exactly two independent meshes, regardless of resistances or voltage sources.

Practical application and limitations

Mesh analysis works cleanly for circuits containing only voltage sources and resistors, or when current sources lie on the perimeter. A current source shared between two meshes complicates the setup because a single mesh current cannot flow through it; you must either treat it as a constraint or convert it to an equivalent voltage source. For circuits with many current sources or nonplanar topology, nodal analysis often proves simpler.

The term reflects the visual appearance of the method: when you draw a circuit and mark each independent loop, the loops interlace like threads in a mesh fabric. In transformer circuits and three-phase power distribution analysis, mesh currents represent the actual circulating currents that flow between coupled windings or between phase legs, making the method particularly natural for those applications.