mesh analysis

Mesh analysis is the loop-flavoured twin of nodal analysis. Instead of solving for node voltages using the current law, it solves for loop currents using the voltage law. A mesh is one of the open windows of a flat circuit drawing, a loop with no other loops inside it. The trick is to imagine a current circulating around each window, and then let the algebra figure out how big each circulating current must be.

The procedure mirrors nodal analysis. Assign a circulating current, usually clockwise, to each mesh. Then walk Kirchhoff's voltage law around each mesh, writing each resistor's voltage drop as its resistance times the net current through it, which is the difference of the two mesh currents where two meshes share a resistor. That gives one equation per mesh, and you solve the set. A two-mesh circuit yields two equations in two unknown loop currents, and once you have them, every branch current is a sum or difference of the loop currents.

Mesh analysis is often the quickest hand method for circuits built from many series loops and voltage sources, complementing nodal analysis, which is friendlier when there are many parallel branches and current sources. One honest restriction: the classic mesh method assumes the circuit is planar, meaning you can draw it flat with no wires crossing over one another. Nodal analysis has no such limitation, so for tangled non-planar circuits, reach for nodal instead.