Question

Determine the current in each branch of the network shown in figure:

Answer 1

Current flowing through various branches of the circuit is represented in the given figure. 

I₁ = Current flowing through the outer circuit
I₂ = Current flowing through branch AB
I₃ = Current flowing through branch AD
I₂ − I₄ = Current flowing through branch BC
I₃ + I₄ = Current flowing through branch CD
I₄ = Current flowing through branch BD

For the closed circuit ABDA, potential is zero i.e.,
10I₂ + 5I₄ − 5I₃ = 0
2I₂ + I₄ −I₃ = 0
I₃ = 2I₂ + I₄ … (1)

For the closed circuit BCDB, potential is zero i.e.,
5(I₂ − I₄) − 10(I₃ + I₄) − 5I₄ = 0
5I₂ + 5I₄ − 10I₃ − 10I₄ − 5I₄ = 0
5I₂ − 10I₃ − 20I₄ = 0
I₂ = 2I₃ + 4I₄ … (2)

For the closed circuit ABCFEA, potential is zero i.e.,
−10 + 10 (I₁) + 10(I₂) + 5(I₂ − I₄) = 0
10 = 15I₂ + 10I₁ − 5I₄
3I₂ + 2I₁ − I₄ = 2 … (3)

From equations (1) and (2), we obtain 
I₃ = 2(2I₃ + 4I₄) + I₄ 
I₃ = 4I₃ + 8I₄ + I₄
− 3I₃ = 9I₄
− 3I₄ = + I₃ … (4)

Putting equation (4) in equation (1), we obtain 
I₃ = 2I₂ + I₄
− 4I₄ = 2I₂
I₂ = − 2I₄ … (5)

It is evident from the given figure that,
I₁ = I3 + I₂ … (6)

Putting equation (6) in equation (1), we obtain
3I₂ +2(I₃ + I₂) − I₄ = 2
5I₂ + 2I₃ − I₄ = 2 … (7)

Putting equations (4) and (5) in equation (7), we obtain
5(−2 I₄) + 2(− 3 I₄) − I₄ = 2
− 10I₄ − 6I₄ − I₄ = 2
17I₄ = − 2
I₄ =\( \frac{-2}{17} A\)

Equation (4) reduces to
I₃ = − 3(I₄)
I₃ = \(-3(\frac{-2}{17}) = \frac{6}{17} A\)
I₂ = -2(I₄)
I₂ = \(-2(\frac{-2}{17}) =\frac{ 4}{17} A\)
I₂-I₄ = \(\frac{4}{17} - (\frac{-2}{17}) =\frac{ 6}{17} A\)
I₃+I₄ = \(\frac{6}{17}+(\frac{-2}{17}) =\frac{ 4}{17} A\)
I₁= I₃ +I₂
I₁ = \(\frac{6}{17} +\frac{ 4}{17} = \frac{10}{17} A\)

 

Therefore, 
current in branch AB =\( \frac{4}{17} A\)
current in branch BC = \(\frac{6}{17} A\)
current in branch CD = (\(\frac{-4}{17}\)) A
current in branch AD = \(\frac{6}{17} A\)
current in branch BD = (\(\frac{-2}{17}\)) A

Total Current =\( \frac{4}{17} + \frac{6}{17} +  (\frac{-4}{17}) + \frac{6}{17} + (\frac{-2}{17})\)
\(= \frac{10}{17} A\)