Math Problem Statement

Using Nodal Analysis, determine V1 , V2 and V3

Using Nodal Analysis, determine V1, V2, and V3.

Solution

To solve for V1V_1, V2V_2, and V3V_3 using nodal analysis, let's proceed by applying Kirchhoff's Current Law (KCL) at each node.

Given Values:

  • Resistances:
    • R1=10ΩR_1 = 10 \Omega
    • R2=5ΩR_2 = 5 \Omega
    • R3=20ΩR_3 = 20 \Omega
    • R4=10ΩR_4 = 10 \Omega
    • R5=5ΩR_5 = 5 \Omega
  • Current source: I=10 AI = 10 \text{ A}

Step-by-Step Solution:

1. Define the Nodes

  • Node V1V_1 is the top-left corner.
  • Node V2V_2 is the central node where the current source connects.
  • Node V3V_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 V2V_2, we will take it into account directly in the equation for V2V_2.

Node V1V_1:

Using KCL, the current going out of node V1V_1 must equal zero: V1V25+V1010+V1V320=0\frac{V_1 - V_2}{5} + \frac{V_1 - 0}{10} + \frac{V_1 - V_3}{20} = 0

Node V2V_2:

The current source enters node V2V_2, so we have: V2V15+V2V310=10\frac{V_2 - V_1}{5} + \frac{V_2 - V_3}{10} = 10

Node V3V_3:

Using KCL at node V3V_3, we get: V3V120+V3V210+V305=0\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. V1V25+V110+V1V320=0\frac{V_1 - V_2}{5} + \frac{V_1}{10} + \frac{V_1 - V_3}{20} = 0
  2. V2V15+V2V310=10\frac{V_2 - V_1}{5} + \frac{V_2 - V_3}{10} = 10
  3. V3V120+V3V210+V35=0\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 V1V_1, V2V_2, and V3V_3 are as follows:

  • V1=5001145.45VV_1 = \frac{500}{11} \approx 45.45 \, \text{V}
  • V2=8001172.73VV_2 = \frac{800}{11} \approx 72.73 \, \text{V}
  • V3=3001127.27VV_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 V1V_1, V2V_2, and V3V_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 V1V_1 and V3V_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.

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Math Problem Analysis

Mathematical Concepts

Electrical Engineering
Circuit Analysis
Nodal Analysis
Kirchhoff's Current Law (KCL)

Formulas

KCL: Sum of currents entering a node equals the sum of currents leaving the node
Ohm's Law: V = IR

Theorems

Kirchhoff's Current Law (KCL)
Ohm's Law

Suitable Grade Level

College (Undergraduate Electrical Engineering)