Mesh-Current Analysis & Supermeshes

Mesh analysis reduces a planar circuit to one unknown current per elementary face. It is a KVL formulation: each mesh equation sums the voltage drops produced by its current and by neighboring mesh currents in shared branches.

The impedance matrix #

Assign every mesh current clockwise. For resistor-only meshes, the matrix has the pattern

$$ZI=V,$$

where

• $Z_{kk}$ is the sum of resistances around mesh $k$;
• $Z_{kj}$ is minus the resistance shared by meshes $k$ and $j$;
• $V_k$ is the signed sum of source rises around mesh $k$.

If a branch belongs only to mesh $k$, its branch current is $I_k$. If shared by meshes $k$ and $j$ whose references oppose in that branch, the branch current is $I_k-I_j$.

Current sources #

An independent current source on the outer boundary sets one mesh current directly. A current source shared by two meshes does not provide its voltage, so form a supermesh around both adjacent meshes:

1. omit the current-source branch from the KVL path;
2. write KVL around the combined perimeter;
3. add the constraint $I_k-I_j=I_s$ with the correct sign.

Dependent current sources add the corresponding controlling relation.

When not to use it #

Mesh analysis requires a planar representation. A general loop-current method exists for nonplanar graphs, but the simple "one current per window" procedure does not. Nodal analysis is usually smaller when there are many current sources or fewer essential nodes than meshes.

Verify the solution by reconstructing branch currents, checking KCL at nodes, and confirming total absorbed and delivered power balance.

The symmetric off-diagonal entries are not a coincidence: each is minus the shared resistance for this passive reciprocal network. An asymmetric hand-built mesh matrix is therefore an immediate transcription warning.