Math Problem Statement
Solution
We are tasked with calculating the current labeled using the superposition theorem for the given circuit. The labeled quantities are:
Applying Superposition Theorem
The superposition theorem states that in a linear circuit with multiple independent sources, the total current or voltage for any element can be found by summing the individual contributions from each independent source, with all other sources replaced by their internal resistances (i.e., voltage sources are short-circuited and current sources are open-circuited).
Step 1: Contribution of the Voltage Source ()
- Set the current source to 0 A (open circuit).
- The circuit becomes a simple series circuit with , , and .
The total resistance in the series is:
Using Ohm's Law, the current through the series is:
Since and are in series, this is the current through both resistors. The current through due to is:
Step 2: Contribution of the Current Source ()
- Set the voltage source to 0 V (short circuit).
- The circuit now consists of and in parallel with .
The equivalent resistance of and in parallel is: Thus,
The current divides between and . Using the current division rule, the current through is:
This current is also downward through .
Step 3: Total Current through
Now, we sum the contributions from both sources:
Final Answer:
The current through (downward) is .
Would you like more details on any of the steps?
Here are 5 related questions you might find interesting:
- How would the result change if or were different values?
- How do you calculate current division in a parallel circuit?
- What would happen if or were reversed in polarity?
- How does superposition apply to circuits with dependent sources?
- Can superposition be used for circuits with nonlinear components?
Tip: Always remember to turn off all but one independent source at a time when using superposition.
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Math Problem Analysis
Mathematical Concepts
Superposition Theorem
Ohm's Law
Current Division
Series and Parallel Circuits
Formulas
Ohm's Law: I = V / R
Total resistance in series: R_total = R1 + R2
Equivalent resistance in parallel: 1 / Req = 1 / R1 + 1 / R2
Current division rule: I_R = I_total * (R_other / (R1 + R2))
Theorems
Superposition Theorem
Ohm's Law
Suitable Grade Level
Undergraduate Electrical Engineering
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