# Find Io in the network in Fig. P3.8

Find Io in the network in Fig. P3.8 using nodal analysis.

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

#### Solution:

Let us first “bend” the wires in the circuit diagram to see how the nodes connect.

Notice how we simply moved the resistor connection to the node as it was an “empty” wire. This is the same as the original circuit diagram, however, now, it is much easier for us to see the two main nodes.

Let us label the nodes and the currents as follows:

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

We can now write a KCL equation for the orange node, $V_1$.

$I_1+I_2+4m=2m$

Expressing these currents in terms of voltage and resistance using $I=\dfrac{V}{R}$ gives us:

$\dfrac{V_1}{1k}+\dfrac{V_1-V_2}{2k}+4m=2m\,\,\,\color{orange} {\text{(eq.1)}}$

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

$I_3+I_4=4m+6m$

Again, expressing these currents in terms of voltage and resistance gives us:

$\dfrac{V_2-V_1}{2k}+\dfrac{V_2}{1k}=10m\,\,\,\color{purple} {\text{(eq.2)}}$

Solving equations 1 and 2 simultaneously, we get :

$V_1=1$ v

$V_2=7$ v

From our diagram, we know that $I_0=I_4$.

$I_0=\dfrac{V_2}{1k}$

$I_0=\dfrac{7}{1k}$

$I_0=7$ mA

$I_0=7$ mA