Electrical Circuit Analysis: Simple Techniques and Superposition

Overview

• Discussing methods to solve electrical circuits using techniques like KVL (Kirchhoff's Voltage Law) and KCL (Kirchhoff's Current Law)
• Focus on meshing analysis, nodal analysis, and superposition
• Two simple techniques for solving electrical networks
• Practical application of techniques with problems

Basic Circuit Analysis

• Example Circuit: Voltage source (1V) with two resistors (2kΩ each)
  • Objective: Find voltage across the bottom resistor (VX)
  • Using mesh analysis:
    • Current (I) through the loop: I = Voltage / Total Resistance, I = 1V / 4kΩ = 0.25mA
    • Voltage across one resistor (VX): VX = IR, VX = 0.25mA * 2kΩ = 0.5V

Generalizing the Technique

• Abstract Circuit: Voltage source (V1) with resistors (R1, R2)
  • Objective: Find the voltage across R2 (VX)
  • Using mesh analysis:
    • Current (I) through the loop: I = V1 / (R1 + R2)
    • Voltage across R2 (VX): VX = (R2 / (R1 + R2)) * V1
    • Similarly, voltage across R1 (VY): VY = (R1 / (R1 + R2)) * V1
    • Verification: V1 = VX + VY

Extended Technique with Multiple Resistors

• Circuit with Three Resistors (R1, R2, R3):
  • Voltage across R2: VX = (R2 / (R1 + R2 + R3)) * V1
  • Pattern: Voltage across any resistor in series = (Value of resistor / Sum of all resistors) * Total voltage

Applying the Technique to Current Sources

• Current Source: Current source (I) with resistors in parallel
  • Example with 2 resistors (R1, R2)
  • Using KCL and voltage equality:
    • Total current: I = I1 + I2
    • Voltage across R1 and R2 is the same: I1R1 = I2R2
    • Solving for currents:
      • I1 = (R2 / (R1 + R2)) * I
      • I2 = (R1 / (R1 + R2)) * I
  • Observation: Current through a resistor depends on the other resistor's value

Equivalent Circuits

• Voltage Source with Series Resistance:
  • Adding a resistance in series with a voltage source doesn’t affect the network voltage
• Current Source with Parallel Resistance:
  • Adding a resistance in parallel with a current source doesn’t affect the network current

Key Observations

• Properties of voltage sources:
  • Can supply any current required without changing the voltage
• Properties of current sources:
  • Can maintain a constant current regardless of the voltage across parallel elements
  • Useful for simplifying circuit networks