https://file.aiquickdraw.com/mathbot/file/1726083288622nmwo06pf.jpg

To solve the problem using Nodal Analysis, we need to determine the voltages at the nodes and then calculate the current flowing through each resistor. Here’s the step-by-step approach:

### Step 1: Identify Nodes and Assign Voltages
1. Let \( V_1 \) be the voltage at the first node (between the 5 Ω and 15 Ω resistors).
2. Let \( V_2 \) be the voltage at the second node (between the 10 Ω and 8 Ω resistors).
3. Ground the bottom node to be 0 V (reference node).

### Step 2: Apply KCL (Kirchhoff’s Current Law) at Each Node
Let's set up the equations by applying KCL at nodes \( V_1 \) and \( V_2 \).

#### Node \( V_1 \):
The sum of currents leaving node \( V_1 \) is zero:

\[
\frac{V_1 - 4}{5} + \frac{V_1 - V_2}{10} + \frac{V_1}{15} = 0
\]

Simplify this equation:

\[
\frac{V_1 - 4}{5} + \frac{V_1 - V_2}{10} + \frac{V_1}{15} = 0
\]

Multiply by 30 to clear the fractions:

\[
6(V_1 - 4) + 3(V_1 - V_2) + 2V_1 = 0
\]

\[
6V_1 - 24 + 3V_1 - 3V_2 + 2V_1 = 0
\]

\[
11V_1 - 3V_2 = 24  \quad \text{(Equation 1)}
\]

#### Node \( V_2 \):
The sum of currents leaving node \( V_2 \) is zero:

\[
\frac{V_2 - V_1}{10} + \frac{V_2}{8} + \frac{V_2 - 6}{12} = 0
\]

Multiply by 120 to clear the fractions:

\[
12(V_2 - V_1) + 15V_2 + 10(V_2 - 6) = 0
\]

\[
12V_2 - 12V_1 + 15V_2 + 10V_2 - 60 = 0
\]

\[
37V_2 - 12V_1 = 60 \quad \text{(Equation 2)}
\]

### Step 3: Solve the System of Equations
Now, solve the system of equations:

\[
\begin{cases}
11V_1 - 3V_2 = 24 \\
-12V_1 + 37V_2 = 60
\end{cases}
\]

Let's solve these equations.The solutions for the node voltages are:

\[
V_1 = \frac{1068}{371} \approx 2.88 \, \text{V}
\]

\[
V_2 = \frac{948}{371} \approx 2.55 \, \text{V}
\]

### Step 4: Find the Current Through Each Resistor
Now, calculate the currents through each resistor using Ohm’s law.

1. **Current through 5 Ω resistor:**

   \[
   I_{5\Omega} = \frac{V_1 - 4}{5} = \frac{2.88 - 4}{5} = -0.224 \, \text{A} \, (\text{or} \, 224 \, \text{mA}, \text{flowing towards } 4V)
   \]

2. **Current through 10 Ω resistor:**

   \[
   I_{10\Omega} = \frac{V_1 - V_2}{10} = \frac{2.88 - 2.55}{10} = 0.033 \, \text{A} \, (\text{or} \, 33 \, \text{mA})
   \]

3. **Current through 15 Ω resistor:**

   \[
   I_{15\Omega} = \frac{V_1}{15} = \frac{2.88}{15} = 0.192 \, \text{A} \, (\text{or} \, 192 \, \text{mA})
   \]

4. **Current through 8 Ω resistor:**

   \[
   I_{8\Omega} = \frac{V_2}{8} = \frac{2.55}{8} = 0.319 \, \text{A} \, (\text{or} \, 319 \, \text{mA})
   \]

5. **Current through 12 Ω resistor:**

   \[
   I_{12\Omega} = \frac{V_2 - 6}{12} = \frac{2.55 - 6}{12} = -0.288 \, \text{A} \, (\text{or} \, 288 \, \text{mA}, \text{flowing towards } 6V)
   \]

Would you like more details or have any questions? Here are some related questions:

1. How is nodal analysis different from mesh analysis?
2. What would happen if we changed one of the voltage sources?
3. How does the polarity of a voltage source affect current direction?
4. What other methods can be used to solve circuit problems besides nodal analysis?
5. How can you verify your results using simulation software?

**Tip:** Always double-check the direction of the currents when using nodal analysis, as incorrect assumptions can lead to sign errors.

Find the current flowing through all resistors using Nodal Analysis

Learn how to apply Nodal Analysis to find the current through all resistors in a circuit. This problem involves Kirchhoff's Current Law (KCL), Ohm's Law, and solving a system of equations. Suitable for university-level electrical engineering students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation