Superposition Theorem Lecture Notes

Introduction to Superposition Theorem

• Definition: The superposition theorem states that the voltage across or the current through an element in a linear circuit is the algebraic sum of the voltages across or the currents through that element due to each independent source acting alone.
• Context: Useful for analyzing linear bidirectional networks with multiple independent sources.

Key Points

• Turning Off Sources:

  • In a linear bidirectional network containing more than one independent source, the response in any element is the sum of the responses obtained with one source acting at a time and the other sources being turned off.
  • Turning Off Definition:
    • Replace every voltage source with 0 volts (short circuit).
    • Replace every current source with 0 amperes (open circuit).

• Important Reminders:

  • Do not turn off dependent sources; they remain as they are.
  • The superposition theorem is not valid for non-linear circuits.

Example Problem

• Objective: Find voltage across a 4 ohm resistor using superposition theorem.

Given Network

• Contains one voltage source (6V) and one current source (3A).

Step 1: Analyze with 6V Source

1. Turn off the 3A current source (open circuit).
2. New notation for voltage across the resistor: V1.
3. Apply Kirchhoff's Voltage Law (KVL):
   • Equation: +6V - I1 × 8Ω - I1 × 4Ω = 0
   • Solve for I1:
     • I1 = 6 / (8 + 4) = 0.5A
4. Calculate V1:
   • V1 = I1 × 4Ω = 0.5 × 4 = 2V

Step 2: Analyze with 3A Source

1. Turn off the 6V source (short circuit).
2. New notation for voltage across the resistor: V2.
3. Use the current divider rule to find I2:
   • I2 = (Total Current) × (Other Resistance) / (Sum of Resistances)
   • I2 = 3 × 8 / (8 + 4) = 2A
4. Calculate V2:
   • V2 = I2 × 4Ω = 2 × 4 = 8V

Final Calculation

• Net Voltage: V = V1 + V2 = 2V + 8V = 10V

Conclusion

• The voltage across the 4 ohm resistor using the superposition theorem is 10 volts.