Kirchhoff's Current Law (KCL) Recap

Overview of KCL

• KCL states that the algebraic sum of currents entering a node is zero.
• This law is fundamental in circuit analysis.

Definitions

• Node: A point where two or more circuit elements meet.
• Currents:
  • Let i1, i2, i3, and i4 be currents in branches connected to node A.
  • Assume:
    • Positive sign for currents entering a node.
    • Negative sign for currents exiting a node.

Mathematical Representation

• KCL can be expressed as:
  i1 - i2 - i3 + i4 = 0
  or
  i1 + i4 = i2 + i3
• This represents conservation of charge: currents entering a node equal currents exiting.

Nodal Analysis

• Nodal analysis finds currents using node voltages.
• Node Voltage: Voltage of a node with respect to ground.

Current through Resistors

• Current flowing through a resistor is determined by the potential difference across it:
  • i1 = (V1 - Va) / R1
  • i2 = (Va - V2) / R2
  • i3 = (Va - V3) / R3
  • i4 = (V4 - Va) / R4

Example Problem

1. Assign node voltages.
2. Use KCL to find relationships between the currents.
3. Substitute current equations into KCL expression.
4. Solve for node voltages and subsequently the currents.
5. A negative current value indicates the assumed direction was incorrect.

Sample Calculation Steps

• Assign node A voltage Va and node B voltage Vb (grounded so Vb = 0).
• Assume current directions (e.g., i1, i2, i3).
• Apply KCL:
  • i1 + i2 + i3 = 0
• Substitute current equations based on voltages and resistances.
• Solve for voltages and derive currents.

Additional Example with Current Sources

• Question: Calculate current through an 8-ohm resistor with a voltage and current source.
• Assign node voltages and assume current directions.
• Apply KCL:
  • i1 + i2 = i3
• Given an ideal current source, i1 = 1 A.
• Solve for other currents using KCL and voltage relationships.

Conclusion

• KCL is essential for analyzing circuits with both voltage and current sources.
• Understanding node voltages and current assumptions is key to solving circuit problems effectively.