In the circuit given below, R = 15 Ω. Find vo using mesh analysis.?

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  • 1 year ago
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    To write mesh equations add all the resistance in a loop, change to negative

    giving -30i1. The i2 resistance is the resistor common to loops 1 and 2 which is 20 in this case. Assume clockwise currents i1 and i2 in left and right loops. Equation for left loop:

    -30i1+20i2-60=0 (-60 because that is a voltage drop in clockwise direction).

    20i2 = + 30i1 + 60

    i2 = (1.5i1 + 3)

    Notice the dependent source 5vo = 5*(-4i1) = -20i1

    Equation of the right loop

    20i1 - 35i2 + 60 - 5Vo = 0 but 5Vo = -20i1 and -5Vo = 20i1

    Giving:

    20i1 - 35i2 +60 +20i1 = 0

    But i2 = (1.5i1 + 3) from above

    Substitute in

    20i1 -35(1.5i1 + 3) + 60 + 20i1 = 0

    20i1 -52.5i1 -105 + 60 + 20i1 = 0

    -12.5i1 - 45 = 0-----> i1 = 45/(-12.5) = -3.6

    Vo = -4i1 = -4*(-3.6) = 14.4Volts

    i2 = 1.5(-3.6) + 3 = -2.4

    Verify by comparing power IN from sources to power consumed by resistance:

    10Ω*3.6²+20Ω*1.2²+15Ω*2.4² = 244.8W by resistance

    60V*1.2A+72V*2.4A = 244.8W from sources Checks

    i1 = 3.6A counter clock wise

    i2 = 2.4 counter clock wise

    i1-i2 = 1.2A up the middle branch

    This is the correct solution. Please award best answer to the Best Answer

    Do Node solution real quick. Lev V be node above +60V source. Write KCL of currents leaving V assuming bottom of ckt is ground

    V/10+(V-60)/20+(V-5Vo)/15 = 0

    But Vo = V/10 *4 = 0.4V so 5Vo = 2V

    V/10+(V-60)/20+(V-2V)/15 = 0

    Multiply equation by 60

    6V+3V-180+4V-8V= 0

    5V = 180 so V = 180/5 = 36V

    i1 = V/10 = 36/10 = 3.6 down into 6Ω

    (36-60)/20 = -1.2 or 1.2A up out of the 60V source

    (V-5Vo)/15 = -V/15 = -36/15 = -2.4 or 2.4 up out of 5Vo source

    Same answers. Please ask questions if not clear.

  • Roger
    Lv 7
    1 year ago

    Vo = 14.4 volts

    First Mesh equation ( the left loop)

    4 (I1) + 6(I1) + 20( I1-I2) + 60 volts = 0

    note the + end of the 4 ohm resistor is on the bottom, the + end of the 6 ohm resistor is on the left, and the + end of the 20 ohm resistor is on the top

    30(I1) - 20(I2) = -60 volts

    divide by 10

    3I1 - 2I2 = -6

    second mesh equation ( right loop)

    (I2)15 +5vo -20(I1 -I2)- 60V =0

    but vo = -4I1

    15(I2)-20(I1) -20(I1 -I2) = 60 volts

    60 volts = -40(I1) +35(I2)

    divide by 5

    12 volts = -8I1 +7I2

    3I1 - 2I2 = -6 volts

    solve for I1 and I2

    I1 = -3.6 amps

    I2 = - 2.4 amos

    Vo = - 4(-3.6 amps) = 14.4 volts

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