Why mesh analysis

When a circuit has several loops, guessing branch currents gets messy, mesh analysis makes it mechanical.

• A mesh is a loop with no smaller loop inside it, the circuit's smallest windows.
• You assign one mesh current per mesh (all clockwise, by convention) and solve for those instead of every branch current.
• Once the mesh currents are known, any branch current is a mesh current or the difference of two.

Writing the equations

The whole method is just KVL applied once per mesh, with Ohm's law for every resistor.

• Travel around each mesh in the direction of its current and add the voltage drops; set the sum to zero (∑ v = 0).
• A resistor inside one mesh drops R·iₖ. A resistor shared between two meshes drops R(iₖ − iⱼ), the difference of the two mesh currents.
• A source is positive on the right side when it aids the mesh current. Getting the source signs right is where care pays off.

Solving and checking

Two meshes give two equations; three give three, a small linear system.

• For two meshes the equations take the form R₁₁i₁ − Rₘi₂ = V₁ and −Rₘi₁ + R₂₂i₂ = V₂, where R₁₁ is the total resistance in mesh 1 and Rₘ is the shared resistor.
• Solve by substitution or Cramer's rule; the two-mesh solver on this deck does it live once you enter the self- and mutual-resistances.
• Always sanity-check: a mesh current can come out negative, which simply means the real current flows opposite to your assumed direction.