calculate Va-Vd for the circuit given above.?

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  • 1 month ago

    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

  • Ash
    Lv 7
    1 month ago

    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

     

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