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24/257 A\text{24/257 A}

Step-by-step solution

4 steps
  1. Simplify the circuit using source transformation.

    First, convert the 5A5\text{A} current source on the left with its parallel 6Ω6\Omega resistor into an equivalent voltage source.
    The voltage source value will be VL=I×R=5A×6Ω=30VV_L = I \times R = 5\text{A} \times 6\Omega = 30\text{V}.
    Since the current source points downwards, the positive terminal of the voltage source will be at the bottom (towards ground), and the negative terminal will be towards node NLN_L. So, it contributes −30V-30\text{V} to node NLN_L. This 30V30\text{V} source will be in series with the 6Ω6\Omega resistor.

    Similarly, convert the 3A3\text{A} current source on the right with its parallel 4Ω4\Omega resistor into an equivalent voltage source.
    The voltage source value will be VR=I×R=3A×4Ω=12VV_R = I \times R = 3\text{A} \times 4\Omega = 12\text{V}.
    Since the current source points upwards, the positive terminal of the voltage source will be towards node NRN_R, and the negative terminal will be at the bottom (towards ground). So, it contributes +12V+12\text{V} to node NRN_R. This 12V12\text{V} source will be in series with the 4Ω4\Omega resistor.

    The circuit now looks like this, with two main nodes NLN_L and NRN_R (assuming the bottom wire is ground 0V0\text{V}):

    • From node NLN_L to ground: A 3Ω3\Omega resistor in parallel with a series combination of a 6Ω6\Omega resistor and a −30V-30\text{V} voltage source.
    • From node NRN_R to ground: A 3Ω3\Omega resistor in parallel with a series combination of a 4Ω4\Omega resistor and a +12V+12\text{V} voltage source.
    • Between nodes NLN_L and NRN_R: A 7Ω7\Omega resistor (through which ioi_o flows), and a series combination of a 1Ω1\Omega resistor and a 5V5\text{V} voltage source. The 5V5\text{V} source has its positive terminal towards NLN_L (through the 1Ω1\Omega resistor) and its negative terminal towards NRN_R.
  2. Apply Kirchhoff's Current Law (KCL) at nodes NLN_L and NRN_R using nodal analysis.

    KCL at node NLN_L (sum of currents leaving the node is zero):

    NL−NR7Ω+NL−(NR+5V)1Ω+NL3Ω+NL−(−30V)6Ω=0\frac{N_L - N_R}{7\Omega} + \frac{N_L - (N_R + 5\text{V})}{1\Omega} + \frac{N_L}{3\Omega} + \frac{N_L - (-30\text{V})}{6\Omega} = 0 NL−NR7+(NL−NR−5)+NL3+NL+306=0\frac{N_L - N_R}{7} + (N_L - N_R - 5) + \frac{N_L}{3} + \frac{N_L + 30}{6} = 0

    Multiply the equation by 42 (the least common multiple of 7,1,3,67, 1, 3, 6) to eliminate fractions:

    6(NL−NR)+42(NL−NR−5)+14NL+7(NL+30)=06(N_L - N_R) + 42(N_L - N_R - 5) + 14N_L + 7(N_L + 30) = 0 6NL−6NR+42NL−42NR−210+14NL+7NL+210=06N_L - 6N_R + 42N_L - 42N_R - 210 + 14N_L + 7N_L + 210 = 0 (6+42+14+7)NL+(−6−42)NR=0(6 + 42 + 14 + 7)N_L + (-6 - 42)N_R = 0 69NL−48NR=0(Equation A)69N_L - 48N_R = 0 \quad (\text{Equation A})

    KCL at node NRN_R (sum of currents leaving the node is zero):

    NR−NL7Ω+NR−(NL−5V)1Ω+NR3Ω+NR−12V4Ω=0\frac{N_R - N_L}{7\Omega} + \frac{N_R - (N_L - 5\text{V})}{1\Omega} + \frac{N_R}{3\Omega} + \frac{N_R - 12\text{V}}{4\Omega} = 0 NR−NL7+(NR−NL+5)+NR3+NR−124=0\frac{N_R - N_L}{7} + (N_R - N_L + 5) + \frac{N_R}{3} + \frac{N_R - 12}{4} = 0

    Multiply the equation by 84 (the least common multiple of 7,1,3,47, 1, 3, 4) to eliminate fractions:

    12(NR−NL)+84(NR−NL+5)+28NR+21(NR−12)=012(N_R - N_L) + 84(N_R - N_L + 5) + 28N_R + 21(N_R - 12) = 0 12NR−12NL+84NR−84NL+420+28NR+21NR−252=012N_R - 12N_L + 84N_R - 84N_L + 420 + 28N_R + 21N_R - 252 = 0 (−12−84)NL+(12+84+28+21)NR+420−252=0(-12 - 84)N_L + (12 + 84 + 28 + 21)N_R + 420 - 252 = 0 −96NL+145NR+168=0-96N_L + 145N_R + 168 = 0 −96NL+145NR=−168(Equation B)-96N_L + 145N_R = -168 \quad (\text{Equation B})
  3. Solve the system of equations A and B.

    From Equation A: 69NL=48NR  ⟹  NL=4869NR=1623NR69N_L = 48N_R \implies N_L = \frac{48}{69}N_R = \frac{16}{23}N_R.

    Substitute the expression for NLN_L into Equation B:

    −96(1623NR)+145NR=−168-96\left(\frac{16}{23}N_R\right) + 145N_R = -168 −153623NR+145NR=−168-\frac{1536}{23}N_R + 145N_R = -168

    To combine the terms with NRN_R, find a common denominator:

    −1536NR+145×23NR23=−168\frac{-1536N_R + 145 \times 23 N_R}{23} = -168 −1536NR+3335NR23=−168\frac{-1536N_R + 3335N_R}{23} = -168 1799NR23=−168\frac{1799N_R}{23} = -168 1799NR=−168×231799N_R = -168 \times 23 1799NR=−38641799N_R = -3864 NR=−38641799VN_R = -\frac{3864}{1799}\text{V}

    Now, calculate NLN_L:

    NL=1623NR=1623×(−38641799)N_L = \frac{16}{23}N_R = \frac{16}{23} \times \left(-\frac{3864}{1799}\right) NL=−16×386423×1799=−6182441377VN_L = -\frac{16 \times 3864}{23 \times 1799} = -\frac{61824}{41377}\text{V}

    We can simplify NRN_R and NLN_L by noting that 1799=7×2571799 = 7 \times 257.

    NR=−38647×257=−552257VN_R = -\frac{3864}{7 \times 257} = -\frac{552}{257}\text{V} NL=1623×(−552257)=−16×24257=−384257VN_L = \frac{16}{23} \times \left(-\frac{552}{257}\right) = -\frac{16 \times 24}{257} = -\frac{384}{257}\text{V}
  4. Calculate the current ioi_o through the 7Ω7\Omega resistor.

    The current ioi_o flows from node NLN_L to node NRN_R.

    io=NL−NR7Ωi_o = \frac{N_L - N_R}{7\Omega} io=−384257V−(−552257V)7Ωi_o = \frac{-\frac{384}{257}\text{V} - \left(-\frac{552}{257}\text{V}\right)}{7\Omega} io=−384257+5522577Ai_o = \frac{-\frac{384}{257} + \frac{552}{257}}{7}\text{A} io=552−3842577Ai_o = \frac{\frac{552 - 384}{257}}{7}\text{A} io=1682577Ai_o = \frac{\frac{168}{257}}{7}\text{A} io=168257×7Ai_o = \frac{168}{257 \times 7}\text{A} io=24257Ai_o = \frac{24}{257}\text{A}

    The current ioi_o is 24/257 A\boxed{\text{24/257 A}}.

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