FAD1022 L12 — Kirchhoff's Rules (Applications)

Overview

This lecture applies Kirchhoff's Rules to solve complex multi-loop circuit problems. The focus is on problem-solving strategies, sign conventions, and systematic approaches to analyzing circuits with multiple voltage sources and resistors.

12.1 Kirchhoff's Rules — Problem Solving Strategy

Step-by-Step Approach

  1. Label All Currents

    • Assign current variables ($I_1$, $I_2$, $I_3$...) to each branch
    • Choose directions arbitrarily (if wrong, answer will be negative)

  2. Apply KCL at Junctions

    • Write current conservation equations at nodes
    • Number of independent KCL equations = (number of junctions) − 1

  3. Apply KVL to Loops

    • Choose independent loops
    • Follow sign conventions carefully
    • Number of KVL equations needed = (number of unknown currents) − (number of KCL equations)

  4. Solve Simultaneous Equations

    • Use substitution or elimination methods
    • Check units and signs

Sign Conventions for EMFs

Travel Direction Sign
From (−) to (+) +ℰ
From (+) to (−) −ℰ

Remember: An emf is counted as positive when you traverse it from (−) to (+) terminal, and negative when you go from (+) to (−) terminal.

12.2 Kirchhoff's Rules — Let's Apply It!

Problem #12.2: Multi-Loop Circuit with Switch

Given: Circuit with two batteries and three resistors

  • $12$ V battery, $9$ V battery
  • $7\ \Omega$, $4\ \Omega$, $8\ \Omega$ resistors

Find: Currents $I_1$, $I_2$, and $I_3$ when switch k is closed

Approach:

  1. Apply KCL at junction: $I_1 + I_2 = I_3$
  2. Apply KVL to left loop
  3. Apply KVL to right loop
  4. Solve 3 equations with 3 unknowns

Problem #12.3: Complex Multi-Loop Circuit

Given: Circuit with two batteries and five resistors (Figure 12.3)

  • $24$ V battery, $12$ V battery
  • Resistors: $2\ \Omega$, $4\ \Omega$, $3\ \Omega$, $1\ \Omega$, $5\ \Omega$

Questions: a) Can the circuit be reduced to a single resistor? No — contains multiple voltage sources in different loops

b) Calculate unknown currents $I_1$, $I_2$, and $I_3$

Solution Steps:

Step 1: Apply KCL at node C: $$I_1 = I_2 + I_3 \quad \text{...(1)}$$

Step 2: Apply KVL to Loop 1 (ABCFA): $$-2I_1 - 4I_1 - 3I_3 + 24 = 0$$ $$24 = 6I_1 + 3I_3 \quad \text{...(2)}$$

Step 3: Apply KVL to Loop 2 (FEDCF): $$-12 + 5I_2 + I_2 - 3I_3 = 0$$ $$6I_2 - 3I_3 = 12 \quad \text{...(3)}$$

Step 4: Substitute (1) into (2): $$24 = 6(I_2 + I_3) + 3I_3 = 6I_2 + 9I_3 \quad \text{...(4)}$$

Step 5: Solve equations (3) and (4) simultaneously

Problem #12.5: Multi-Source Circuit (Homework)

Given: Circuit with internal resistances (Figure 12.5)

  • $E_1 = 4$ V, $r_1 = 2\ \Omega$
  • $E_2 = 2$ V, $r_2 = 1\ \Omega$
  • $R = 4\ \Omega$

Key Insight: Using Kirchhoff's current law, label current directions. The current through the middle resistor $R$ is $I_1 + I_2$.

Solution Approach:

  1. Apply KCL at junction
  2. Apply KVL to Loop 1 (left loop)
  3. Apply KVL to Loop 2 (right loop)
  4. Solve for $I_1$ and $I_2$

Mermaid Diagram: Problem Solving Flowchart

flowchart TD
    A[Start Circuit Analysis] --> B[Label All Currents<br/>Choose Directions]
    B --> C[Apply KCL<br/>at Junctions]
    C --> D[Apply KVL<br/>to Loops]
    D --> E{Enough<br/>Equations?}
    E -->|No| D
    E -->|Yes| F[Solve Simultaneous<br/>Equations]
    F --> G[Check Signs<br/>Negative = Reverse Direction]
    G --> H[Verify Conservation<br/>ΣP_in = ΣP_out]
    H --> I[Done!]

Common Mistakes to Avoid

  1. Sign Errors: Be consistent with sign conventions
  2. Current Direction: If answer is negative, direction is opposite — that's okay!
  3. Loop Direction: Can choose clockwise or counterclockwise — just be consistent
  4. Missing Loops: Need enough independent equations for all unknowns

Key Equations Template

For a circuit with $n$ unknown currents:

  • Write $(n-1)$ KCL equations at different junctions
  • Write remaining KVL equations to get $n$ total equations
  • Solve using substitution or matrix methods

Links