Mesh analysis

Mesh analysis solves a planar circuit by assigning a circulating 'mesh current' to each window of the circuit and writing one Kirchhoff-voltage-law equation per window. Rather than tracking the actual current in every wire, you imagine a loop current swirling around each smallest enclosed area; a wire shared by two meshes carries the difference of the two loop currents.

For each mesh you sum the voltage drops around the loop and set them equal to the driving sources, producing as many equations as there are meshes. This is the natural partner to nodal analysis: meshes shine when a circuit has many series loops and few nodes, whereas nodal analysis shines for the opposite. Once you solve for the mesh currents, every real branch current is just a sum or difference of the loop currents threading through it.