Section 4.5: Kirchhoff’s Rules

Kirchhoff’s rules are fundamental tools for analyzing complex electrical circuits. They are based on the conservation of charge and energy.

  • Kirchhoff’s Current Law (KCL): The total current entering a junction equals the total current leaving: \[ \sum I_\text{in} = \sum I_\text{out} \]
  • Kirchhoff’s Voltage Law (KVL): The sum of voltages around any closed loop in a circuit is zero: \[ \sum V = 0 \]
  • KCL and KVL are used together to solve for unknown currents and voltages in circuits with multiple loops and branches.

Example: Two-Loop Circuit

A circuit has two loops with resistors R1 = 4 Ω, R2 = 6 Ω, R3 = 3 Ω and a 12 V battery in loop 1 and 6 V battery in loop 2. Find the currents using Kirchhoff’s rules.

Assign currents I1 in loop 1 and I2 in loop 2. Apply KVL:
Loop 1: \( 12 - 4I_1 - 3(I_1 - I_2) = 0 \)
Loop 2: \( 6 - 6I_2 - 3(I_2 - I_1) = 0 \)
Solve simultaneously:
I1 ≈ 2.14 A, I2 ≈ 0.71 A

Practice Problems

  1. Apply KCL to a junction where 3 currents meet: I1 = 2 A in, I2 = 1.5 A in. Find I3 leaving.
  2. Use KVL in a loop with a 9 V battery and resistors 2 Ω, 3 Ω, 4 Ω to find the current.
  3. A circuit has two loops with resistors 5 Ω, 10 Ω, 2 Ω and batteries 12 V, 6 V. Set up KVL equations.
  4. Explain the physical meaning of Kirchhoff’s laws in terms of charge and energy conservation.
  5. Design a simple two-loop circuit and calculate all branch currents using Kirchhoff’s rules.
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