GATE 2023 Network Theory — KVL Mesh Analysis (Q14 Complete Solution)
Subject: Network Theory
The Question
A circuit has two meshes. Mesh 1 contains a 10V source, a 2Ω resistor, and a 4Ω resistor shared with Mesh 2. Mesh 2 contains the shared 4Ω resistor and a 6Ω resistor. Find the current through the 4Ω resistor.
Concept First — What is Mesh Analysis?
Before jumping to the solution, understand why we use mesh analysis:
- It is a systematic method to find currents in a planar circuit
- We apply KVL (Kirchhoff's Voltage Law) to each mesh
- KVL says: Sum of all voltages around a closed loop = 0
This avoids guessing current directions. You define mesh currents (I₁, I₂) and let the math handle the signs.
💡 Tip
Always define mesh currents in the same direction (usually clockwise). This keeps your equations consistent and avoids sign errors.
Step-by-Step Solution
📌 Step 1: Define mesh currents
Let I₁ = current in Mesh 1 (clockwise)
Let I₂ = current in Mesh 2 (clockwise)
The 4Ω resistor is shared between both meshes. Current through it = I₁ - I₂
📌 Step 2: Apply KVL to Mesh 1
Going clockwise around Mesh 1:
+10 - 2(I₁) - 4(I₁ - I₂) = 0
Expand:
10 - 2I₁ - 4I₁ + 4I₂ = 0
10 - 6I₁ + 4I₂ = 0
Equation 1: 6I₁ - 4I₂ = 10
📌 Step 3: Apply KVL to Mesh 2
Going clockwise around Mesh 2:
-4(I₂ - I₁) - 6(I₂) = 0
Note: No voltage source in Mesh 2.
Expand:
-4I₂ + 4I₁ - 6I₂ = 0
4I₁ - 10I₂ = 0
Equation 2: 4I₁ - 10I₂ = 0
📌 Step 4: Solve the simultaneous equations
From Equation 2:
4I₁ = 10I₂
I₁ = 2.5 I₂
Substitute into Equation 1:
6(2.5 I₂) - 4I₂ = 10
15I₂ - 4I₂ = 10
11I₂ = 10
I₂ = 10/11 ≈ 0.909 A
Then: I₁ = 2.5 × (10/11) = 25/11 ≈ 2.27 A
📌 Step 5: Find current through 4Ω resistor
Current through 4Ω = I₁ - I₂
= 25/11 - 10/11
= 15/11 ≈ 1.36 A
Key Formula Used
Sum of voltages around any closed loop = 0 (KVL)
Final Answer
✅ Final Answer
Current through 4Ω resistor = 15/11 A ≈ 1.36 A
What to Remember for GATE
- Mesh analysis works best when there are fewer meshes than nodes
- If a current source exists between two meshes → use supermesh technique
- For n meshes: you'll write n equations and solve n unknowns
- Practice 3-mesh circuits — they appear frequently in GATE
💡 Tip
In GATE, always check units. Current in Amperes, resistance in Ohms, voltage in Volts. Simple checks save marks.
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