# Find V1 and V2 in the circuit

Find V1 and V2 in the circuit in Fig. P3.5 using nodal analysis.

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

#### Solution:

We will label two nodes to perform nodal analysis on this circuit as follows:

Let us first look at node $V_1$, the orange node.

We can write a KCL expression like so (note that $k=10^3$ and $m=10^{-3}$):

$I_1+I_2+6m=4m$

(remember that while $I_1, I_2$ and the 6mA leave the node, 4mA is coming into the node)

We can express these currents in terms of voltage and resistance using (I=V/R).

$\dfrac{V_1}{6k}+\dfrac{V_1-V_2}{4k}+6m=4m\,\,\,\color{orange} {\text{(eq.1)}}$

Let us now look at node $V_2$, the purple node and write a KCL expression.

$I_3+I_4+I_5=6m$

Again, expressing these currents in terms of voltage and resistance, we get:

$\dfrac{V_2-V_1}{4k}+\dfrac{V_2}{6k}+\dfrac{V_2}{3k}=6m\,\,\,\color{purple} {\text{(eq.2)}}$

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

$V_1=0$ v

$V_2=8$ v

$V_1=0$ v
$V_2=8$ v