Math Problem Statement
Using Nodal Analysis, determine V1 , V2 and V3
Solution
To solve for , , and using nodal analysis, let's proceed by applying Kirchhoff's Current Law (KCL) at each node.
Given Values:
- Resistances:
- Current source:
Step-by-Step Solution:
1. Define the Nodes
- Node is the top-left corner.
- Node is the central node where the current source connects.
- Node 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 , we will take it into account directly in the equation for .
Node :
Using KCL, the current going out of node must equal zero:
Node :
The current source enters node , so we have:
Node :
Using KCL at node , we get:
3. Simplify and Solve the Equations
Now we have the following system of equations:
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 , , and are as follows:
Would you like further details on each step, or do you have any additional questions?
Here are some related questions to deepen your understanding:
- How would the values of , , and change if the current source were reversed?
- What happens to the node voltages if the resistance values are doubled?
- Can we solve this circuit using mesh analysis instead of nodal analysis?
- How would adding another resistor between nodes and affect the circuit?
- 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)
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