calculate Va-Vd for the circuit given above.?

Question

Answer 1

F is ground.

voltages            currents

a  17.794         I1   412.7 mA

b    9.54           I2       1.214 A

c  15.951         I3  -801.336mA

d  28.351

e    4.952

f     0.0

g    2.256

Answer 2

Let node "d" be the ground

Given 𝑖₁ = 𝑖₂ + 𝑖₃ ...............(1)

Then using Mesh Analysis we analyze left loop

18 - 0.5𝑖₁ - 20𝑖₁ - 6𝑖₃ + 3.0 - 0.25𝑖₃ - 15𝑖₁ = 0 

-35.5𝑖₁ - 6.25𝑖₃ + 21 = 0

 3550𝑖₁ + 625𝑖₃ = 2100

142𝑖₁ + 25𝑖₃ = 84 ..............(2)

plug in 𝑖₁ in (2)

142(𝑖₂ + 𝑖₃) + 25𝑖₃ = 84 

142𝑖₂ + 142𝑖₃ + 25𝑖₃ = 84

142𝑖₂ + 167𝑖₃ = 84 ..............(3)

Now using Mesh Analysis for right loop

0.25𝑖₃ - 3 + 6𝑖₃ - 8𝑖₂ + 12 - 0.5𝑖₂ - 0.75𝑖₂ - 24 = 0

-9.25𝑖₂ + 6.25𝑖₃ - 15 = 0

925𝑖₂ - 625𝑖₃ = - 1500

37𝑖₂ - 25𝑖₃ = - 60 ..............(4)

Multiply (2) by 25 and (3) by 167 and then add

3550𝑖₂ + 4175𝑖₃ + 6179𝑖₂ - 4175𝑖₃ = 2100 - 10020

9729𝑖₂ = -7920

𝑖₂ ≈ - 0.814 A .............(5)

Negative sign denotes that 𝑖₂ is flowing in reverse than originally shown.

plug (5) in (4)

 37(-0.814) - 25𝑖₃ = - 60

-30.118 - 25𝑖₃ = - 60

- 25𝑖₃ = -29.882

𝑖₃ ≈ 1.195 A ............(6)

From (1)

 

𝑖₁ = -0.814 +1.195 = 0.381 A ...........(7)

so we get 

𝑖₁ = 0.381 A

𝑖₂ = 0.814 A flowing in opposite direction to shown originally

𝑖₃ = 1.195 A

Since we have node d at ground potential

Vc - Vd = (-0.814 A)(0.5Ω) - 12V = -11.593 V .

Vb - Vc = (-0.814 A)(8Ω) = -6.512 V

Va - Vb = (0.381 A)(20Ω) = 7.62 V

Sum of above gives

(Va - Vb) + (Vb - Vc) + (Vc - Vd) = 7.62 V + (-6.512 V) + (-11.593 V)

Va - Vd = - 10.485 V