Kirchhoff's Rules

Overview

Kirchhoff's Rules (also called Kirchhoff's Laws) are two fundamental principles used to analyze electric circuits. Together with Ohm's Law, they provide a complete framework for solving any DC circuit problem.

Rule	Also Called	Conservation Law	Applies To
KCL	Junction Rule	Charge	Nodes/Junctions
KVL	Loop Rule	Energy	Closed Loops

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Kirchhoff's Current Law (KCL)

Statement: The algebraic sum of currents entering any junction equals zero.

Formula

$$\sum I_{\text{in}} = \sum I_{\text{out}}$$

Or: $$\sum I = 0$$

Key Points

• Based on conservation of charge
• Charge cannot accumulate at a junction
• Also called the Junction Rule

→ See detailed page: Kirchhoff's Current Law (KCL)

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Kirchhoff's Voltage Law (KVL)

Statement: The algebraic sum of all potential differences around any closed loop equals zero.

Formula

$$\sum \Delta V = 0$$

Or: $$\sum \mathcal{E} = \sum IR$$

Key Points

• Based on conservation of energy
• Potential is a state function
• Also called the Loop Rule

→ See detailed page: Kirchhoff's Voltage Law (KVL)

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When to Use Kirchhoff's Rules

Use Kirchhoff's Rules when:

• Circuit has multiple loops
• Circuit has multiple voltage sources
• Resistors cannot be simplified using series/parallel combinations
• You need to find specific branch currents (not just total current)

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Problem-Solving Strategy

Step 1: Label Currents

• Assign a variable to each unknown current
• Choose directions arbitrarily (negative result means opposite direction)

Step 2: Apply KCL

• Write current conservation at junctions
• Need $(n-1)$ equations for $n$ junctions

Step 3: Apply KVL

• Write voltage conservation for independent loops
• Follow sign conventions carefully

Step 4: Solve

• You should have as many equations as unknowns
• Use substitution or matrix methods

Step 5: Check

• Verify power balance: power supplied = power dissipated
• Check that KCL is satisfied at all junctions

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Sign Conventions Summary

Component	Direction	Sign
EMF	(−) → (+)	+ℰ
EMF	(+) → (−)	−ℰ
Resistor	With current	−IR
Resistor	Against current	+IR

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Worked Example: Multi-Loop Circuit

Problem: Find currents in a circuit with two batteries and three resistors.

Solution:

1. Label currents: $I_1$, $I_2$, $I_3$ with assumed directions

2. KCL at junction: $I_1 + I_2 = I_3$

3. KVL Loop 1: $$\mathcal{E}_1 - I_1 R_1 - I_3 R_3 = 0$$

4. KVL Loop 2: $$\mathcal{E}_2 - I_2 R_2 - I_3 R_3 = 0$$

5. Solve: Three equations, three unknowns → solve simultaneously

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Common Applications

• Multi-loop Circuits — complex resistor networks
• Wheatstone Bridge — balanced and unbalanced
• Circuit Analysis — general DC circuit problems
• FAD1022 L11 — Kirchhoff's Rules (Theory) — lecture with fundamentals
• FAD1022 L12 — Kirchhoff's Rules (Applications) — worked examples

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Mermaid Diagram: Kirchhoff's Rules Decision Tree

graph TD
    A[Start Circuit Analysis] --> B{Can simplify<br/>series/parallel?}
    B -->|Yes| C[Use Equivalent<br/>Resistance]
    B -->|No| D[Use Kirchhoff's Rules]
    
    D --> E[Step 1:<br/>Label Currents]
    E --> F[Step 2:<br/>Apply KCL at Junctions]
    F --> G[Step 3:<br/>Apply KVL to Loops]
    G --> H[Step 4:<br/>Solve Equations]
    H --> I[Step 5:<br/>Check Answers]
    
    C --> I
    I --> J[Done!]

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Related Concepts

• Ohm's Law — $V = IR$
• Series and Parallel Circuits — simplification methods
• DC Circuits — direct current circuit analysis
• Electromotive Force (EMF) — voltage sources
• Internal Resistance — real battery behavior