Test: Basic Nodal & Mesh Analysis - 1 - Electrical Engineering (EE) MCQ

Number of junction point for the given circuit is 6.

Using source transformation, the current source is converted to a voltage source as shown below.

Applying nodal analysis at node (x), we get 

Number of loop equation
= b - ( n - 1)
= 8 - ( 4 - 1 ) 
= 8 - 3 = 5

Using source transformation, first we convert the current source into an equivalent voltage source

Applying KVL in loop-1, we get 

Applying KVL in loop-2, we get

On solving equations (i) and (//). we get

i₁ = 2.36 A and i₂ = 4.91 A 

Hence, the current through the battery of 10 V is 

i₂ = 4.91 A

The two mesh equations are: 
5I1 - 3I2 = 10 
-3I1 + 7I2 = -15 
Solving the equations simultaneously, we get I1 = 0.96A and I2 = -1.73A.

To determine the number of Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) equations for the given circuit, we need to analyze its structure. The circuit is a Wheatstone bridge with a voltage source E E E and six resistors (R1 R_1 R1​ to R6 R_6 R6​).

Circuit Analysis:

• Nodes: Identify the distinct nodes where currents split or combine. In a Wheatstone bridge:
  • There is one input node (top) connected to R1 R_1 R1​, R2 R_2 R2​, and R3 R_3 R3​.
  • One output node (bottom) connected to R4 R_4 R4​, R5 R_5 R5​, and R6 R_6 R6​.
  • Two middle nodes where R2 R_2 R2​ and R3 R_3 R3​ meet, and R5 R_5 R5​ and R6 R_6 R6​ meet.
  • Total nodes = 4 (assuming the positive and negative terminals of E E E are separate nodes).
• Loops: A loop is a closed path. The circuit has:
  • One outer loop (through E E E, R1 R_1 R1​, R4 R_4 R4​, and back).
  • Two inner loops (one through R2 R_2 R2​, R6 R_6 R6​, R4 R_4 R4​, and the other through R3 R_3 R3​, R5 R_5 R5​, R4 R_4 R4​).
  • Total independent loops = 3 (using the formula for a planar graph with n n n nodes and b b b branches: L=b−n+1 L = b - n + 1 L=b−n+1, where b=6 b = 6 b=6 branches, n=4 n = 4 n=4 nodes, L=6−4+1=3 L = 6 - 4 + 1 = 3 L=6−4+1=3).

KCL Equations:

• KCL applies at each node, stating that the sum of currents entering equals the sum leaving.
• With 4 nodes, we can write KCL at each. However, one equation is dependent (e.g., the sum of currents at all nodes equals zero due to the circuit being a single connected component). Thus, the number of independent KCL equations is n−1=4−1=3 n - 1 = 4 - 1 = 3 n−1=4−1=3.

KVL Equations:

• KVL applies around each independent loop, stating that the sum of voltage drops equals the source voltage.
• With 3 independent loops, we need 3 KVL equations.

Total Equations:

• Total number of independent equations = KCL + KVL = 3 (KCL) + 3 (KVL) = 6.

However, in circuit analysis, the number of KCL and KVL equations is typically reported as the count of each type needed to solve the circuit, considering independence. For a Wheatstone bridge with one voltage source, the standard approach confirms:

• 3 independent KCL equations (at 3 nodes, excluding one dependent equation).
• 3 independent KVL equations (for 3 loops).

Thus, the number of KVL and KCL equations is 3 each.

Applying KVL in the loop,
5 = 2i — 2i + i or i = 5 A
∴ Value of dependent source
= 2i = 2 x 5 = 10 volt