Circuit Analysis Methods: Mesh analysis

Circuit Analysis Methods: Mesh analysis

1. What Mesh Analysis Is (KVL-based)

Mesh analysis = systematic KVL application to planar circuit's meshes (windows)—loops with no inner loops. Assign mesh current $I_m$ clockwise to each mesh, then write KVL equation per mesh:

where $R_{mm}$ = sum of $R$ in mesh $m$, $R_{mn}$ = $R$ common to meshes $m,n$, $V$ = algebraic sum of voltage sources (positive if rise in mesh direction).

Solve linear system for $I_m$ → branch current = algebraic sum/difference of adjacent mesh currents.

2. Steps (recipe)

1. Identify meshes (e.g., 2 meshes for two-window circuit), assign $I_1,I_2$ clockwise.
2. For each mesh, write KVL: go around mesh, $+R·I_m$ for self, $-R·I_n$ for common $R$ (since neighbour current opposite direction), $+V$ if source from - to + in traversal.
3. Solve simultaneous equations (Cramer's rule, substitution).
4. Branch current = $I_1 - I_2$ on common branch (sign gives direction).

3. Handling Special Cases

• Current source in a mesh: if only one mesh contains current source $I_s$, then $I_m = ±I_s$ directly (sign by direction). If current source shared by two meshes (common branch), create supermesh (combine two meshes, exclude source, write KCL at source: $I_{m1}-I_{m2}=I_s$).

4. Worked Example

Circuit: 10V source in mesh1 with R1=5Ω self, R2=10Ω common with mesh2, mesh2 has  -12V source and R3=8Ω.
Mesh1: (5+10)I1 -10I2 =10
Mesh2: -10I1 + (10+8)I2 = -12   (sign of 12V depends on traversal vs source polarity)
Solve: 15I1-10I2=10, -10I1+18I2=-12 ⇒ I1≈0.176A, I2≈-0.57A (negative means opposite to assumed)
Branch common: I_10Ω = I1 - I2 =0.746A ✓

5. Exam Points

• State mesh analysis is KVL-based with mesh currents — 3-5 marks, frequent.
• Write $R_{mm}I_m - R_{mn}I_n = ΣV$ — 5 marks, frequent.
• Solve 2-mesh circuit numerically — 7-10 marks, very frequent (common mesh numerical).
• Supermesh for shared current source — 5 marks, frequent (often asked).