Circuit Analysis Lecture Notes

Introduction

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Objective

• Find the current through and voltage across every resistor in a given circuit.
• Simplify the circuit into an equivalent circuit with one resistor.

Key Formulas

Series Resistors

• Total Resistance (R_total) = R1 + R2 + R3

Parallel Resistors

• 1/R_total = 1/R1 + 1/R2 + 1/R3

Steps to Simplify the Circuit

1. Identify Series Resistors:

   • Analyze resistors farthest from the battery source.
   • Example: Combine 3 ohms and 12 ohms in series.
     • R_total = 3 + 12 = 15 ohms.

2. Replace Resistors with Equivalent Resistance:

   • Replace the two series resistors with a single equivalent resistor of 15 ohms.

3. Identify Parallel Resistors:

   • Combine two resistors in parallel (e.g., 10 ohms and 15 ohms).
   • Calculation: 1/R_total = 1/10 + 1/15 → R_total = 6 ohms.

4. Continue Simplifying:

   • Combine remaining resistors in series (e.g., 2 ohms, 6 ohms, and 4 ohms).
   • R_total = 2 + 6 + 4 = 12 ohms.

Finding Total Current

• Use Ohm's Law: V = I * R
• Given V = 24 volts and R_total = 12 ohms:
  • I = V/R = 24/12 = 2 amps.

Current Through Resistors in Series

• Current remains constant in series:
  • I1 = I2 = I3 = 2 amps (through all series resistors).

Voltage Across Resistors in Series

• Voltage across resistor = I * R
  • Resistor 1 (2 ohms): V1 = 2 A * 2 ohms = 4 volts.
  • Resistor 2 (6 ohms): V2 = 2 A * 6 ohms = 12 volts.
  • Resistor 3 (4 ohms): V3 = 2 A * 4 ohms = 8 volts.

Recap of Known Values

• Resistor 1: I1 = 2 A, V1 = 4 V.
• Resistor 2: I2 = 2 A, V2 = 12 V.
• Resistor 3: I3 = 2 A, V3 = 8 V.

Voltage at Battery and Across Series Resistors

• Voltage after first resistor: 24 V - 4 V = 20 V.
• Voltage before the last resistor: 0 V + 8 V = 8 V.

Finding Voltage Across Other Resistors

• Resistor 4 and Resistor 5 voltage drops:
  • Voltage across both = 20 V - 8 V = 12 volts.

Current Through Parallel Resistors

• Resistor 4: Current = Voltage/R = 12 V / 10 ohms = 1.2 A.
• Resistor 5: Current = Voltage/R = 12 V / 15 ohms = 0.8 A.

Conclusion and Final Values

• Resistor 4: Voltage = 12 V, Current = 1.2 A.
• Resistor 5: Voltage = 12 V, Current = 0.8 A.
• Resistor 6: Current = 0.8 A, Voltage = 2.4 V.
• Resistor 7: Current = 0.8 A, Voltage = 9.6 V.

Final Thoughts

• All resistor values have been calculated, showing complete analysis of circuit.
• Encouragement to check out further videos for more learning.
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