Math Problem Statement

Using nodal analysis, find the node voltages V1 and V2 in the circuit.

Solution

The problem involves using nodal analysis to find the voltages V1V_1 and V2V_2 in the circuit.

Circuit Information:

  • The circuit has two nodes labeled V1V_1 and V2V_2.
  • There are resistors with values 10Ω10 \, \Omega, 5Ω5 \, \Omega, and 3Ω3 \, \Omega.
  • A 2A2 \, \text{A} current source is connected between node V1V_1 and ground.
  • A voltage source with V2V_2 across it is connected in the circuit.

Steps for Nodal Analysis:

  1. Identify Nodes:

    • V1V_1 is the voltage at the node where the 2A current source is connected.
    • V2V_2 is the voltage at the node with the voltage source connected.
  2. Set Up Equations:

    • Apply Kirchhoff's Current Law (KCL) at nodes V1V_1 and V2V_2.
  3. KCL at Node V1V_1:

    • Assume currents leaving the node.
    • V1V25+V110=2\frac{V_1 - V_2}{5} + \frac{V_1}{10} = 2
  4. KCL at Node V2V_2:

    • V2V15+V23+Isource=0\frac{V_2 - V_1}{5} + \frac{V_2}{3} + I_{\text{source}} = 0

    However, the current source directly gives the current leaving from V1V_1 to ground, so we modify the equation.

  5. Solve the System of Equations:

    • Substitute values and solve for V1V_1 and V2V_2.

Let me calculate the values for V1V_1 and V2V_2 using these steps.The voltages at the nodes are:

  • V1=809V8.89VV_1 = \frac{80}{9} \, \text{V} \approx 8.89 \, \text{V}
  • V2=103V3.33VV_2 = \frac{10}{3} \, \text{V} \approx 3.33 \, \text{V}

These are the node voltages found using nodal analysis.

Would you like further details on the steps or have any questions?

Related Questions:

  1. How would the node voltages change if the 2A current source is replaced with a different value?
  2. What would happen to V1V_1 and V2V_2 if the 3Ω3 \, \Omega resistor is replaced with a 6Ω6 \, \Omega resistor?
  3. How can superposition theorem be applied to this circuit for verification?
  4. What effect would adding another resistor between V2V_2 and ground have on the node voltages?
  5. Can mesh analysis be used as an alternative to find V1V_1 and V2V_2?

Tip:

When solving circuits using nodal analysis, always make sure to carefully account for all the currents entering and leaving each node to avoid errors in setting up your equations.

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Math Problem Analysis

Mathematical Concepts

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

Formulas

Ohm's Law: V = IR
Nodal Analysis Equations: Σ Currents at a Node = 0

Theorems

Kirchhoff's Current Law (KCL)

Suitable Grade Level

Undergraduate Level - Electrical Engineering