To solve for $V_1$, $V_2$, and $V_3$ using nodal analysis, let's proceed by applying Kirchhoff's Current Law (KCL) at each node.

Given Values:

• Resistances:
  • $R_1 = 10 \Omega$
  • $R_2 = 5 \Omega$
  • $R_3 = 20 \Omega$
  • $R_4 = 10 \Omega$
  • $R_5 = 5 \Omega$
• Current source: $I = 10 \text{ A}$

Step-by-Step Solution:

1. Define the Nodes

• Node $V_1$ is the top-left corner.
• Node $V_2$ is the central node where the current source connects.
• Node $V_3$ is the top-right corner.

2. Set Up the Equations at Each Node Using KCL

Since the current source is connected between the reference ground and $V_2$, we will take it into account directly in the equation for $V_2$.

Node $V_1$:

Using KCL, the current going out of node $V_1$ must equal zero: $\frac{V_1 - V_2}{5} + \frac{V_1 - 0}{10} + \frac{V_1 - V_3}{20} = 0$

Node $V_2$:

The current source enters node $V_2$, so we have: $\frac{V_2 - V_1}{5} + \frac{V_2 - V_3}{10} = 10$

Node $V_3$:

Using KCL at node $V_3$, we get: $\frac{V_3 - V_1}{20} + \frac{V_3 - V_2}{10} + \frac{V_3 - 0}{5} = 0$

3. Simplify and Solve the Equations

Now we have the following system of equations:

1. $\frac{V_1 - V_2}{5} + \frac{V_1}{10} + \frac{V_1 - V_3}{20} = 0$
2. $\frac{V_2 - V_1}{5} + \frac{V_2 - V_3}{10} = 10$
3. $\frac{V_3 - V_1}{20} + \frac{V_3 - V_2}{10} + \frac{V_3}{5} = 0$

To make this easier to solve, we can multiply through each equation to eliminate the denominators and then solve the resulting linear system. I will calculate these values now.The solutions for the node voltages $V_1$, $V_2$, and $V_3$ are as follows:

• $V_1 = \frac{500}{11} \approx 45.45 \, \text{V}$
• $V_2 = \frac{800}{11} \approx 72.73 \, \text{V}$
• $V_3 = \frac{300}{11} \approx 27.27 \, \text{V}$

Would you like further details on each step, or do you have any additional questions?

Here are some related questions to deepen your understanding:

1. How would the values of $V_1$, $V_2$, and $V_3$ change if the current source were reversed?
2. What happens to the node voltages if the resistance values are doubled?
3. Can we solve this circuit using mesh analysis instead of nodal analysis?
4. How would adding another resistor between nodes $V_1$ and $V_3$ affect the circuit?
5. What is the total power consumed by this circuit?

Tip: In nodal analysis, always label each node clearly and write KCL equations systematically to avoid errors.