Find Io in the circuit in Fig. P3.10


Find Io in the circuit in Fig. P3.10 using nodal analysis.

Find Io in the circuit in Fig. P3.10

Image from: J. D. Irwin and R. M. Nelms, Basic engineering circuit analysis, 10th ed. Hoboken, NJ: John Wiley, 2011.

Solution:

Show me the final answer↓

To see the nodes clearly, we will “bend” the wires as follows:

Find Io in the circuit in Fig. P3.10

Note that this does not change the original circuit diagram, rather, gives us a new perspective and makes it easier to work with. This can be done as long as there are “empty” wire segments connecting circuit elements together.

Let us now label our nodes and currents. Remember that we must also have a reference ground node, which we chose to be right side of the circuit as shown below.

Find Io in the circuit in Fig. P3.10

In the following equations, k=10^3 and m=10^{-3}

Looking at the orange node, V_1, we will write a KCL equation.

I_1+I_2+2m=1m

We can express these currents in terms of voltage and resistance using I=\dfrac{V}{R}.

\dfrac{V_1-V_2}{2k}+\dfrac{V_1}{8k}+2m=1m\,\,\,\color{orange} {\text{(eq.1)}}

 

Now, we will shift our attention to the purple node, V_2 and write a KCL equation.

I_3+I_4+I_5=2m

Expressing these currents in terms of voltage and resistance gives us:

\dfrac{V_2}{6k}+\dfrac{V_2-V_1}{2k}+\dfrac{V_2}{3k}=2m\,\,\,\color{purple} {\text{(eq.2)}}

 

Solving equations 1 and 2 simultaneously gives us (see full steps):

V_1=0 v

V_2=2 v

 

From our diagram, we know that:

I_0=I_3

I_0=\dfrac{V_2}{6k}

I_0=\dfrac{2}{6k}

I_0=0.33 mA

 

Final Answer:

I_0=0.33 mA

 

This question can be found in Basic Engineering Circuit Analysis, 10th edition, chapter 3, question 3.10.

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