Kirchhoff's Voltage Law (KVL)

Definition

Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all potential differences (voltages) around any closed loop in a circuit is equal to zero.

Or equivalently:

The sum of voltage rises equals the sum of voltage drops around any closed loop.

Formula

$$\sum \Delta V = 0$$

Or:

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

(Sum of emfs equals sum of potential drops across resistors)

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Physical Basis

KVL is based on the conservation of energy:

• Electric potential is a state function
• After traversing a complete loop, you return to the starting potential
• Energy gained from sources equals energy dissipated in resistors

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Sign Conventions

For EMF (Voltage Sources)

Travel Direction	Sign	Reason
From (−) to (+)	+ℰ	Potential rise
From (+) to (−)	−ℰ	Potential drop

For Resistors

Travel Direction	Sign	Reason
Same as current	−IR	Potential drop
Opposite to current	+IR	Potential rise

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How to Apply KVL

Step-by-Step Process

1. Choose a loop: Select a closed path in the circuit
2. Choose direction: Clockwise or counterclockwise
3. Start at any point: Traverse the entire loop
4. Apply sign conventions:
   • +ℰ when going (−) → (+)
   • −ℰ when going (+) → (−)
   • −IR when going with current
   • +IR when going against current
5. Sum to zero: Set algebraic sum equal to zero

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Worked Examples

Example 1: Simple Series Circuit

Circuit: Battery ℰ with resistors $R_1$ and $R_2$ in series

KVL equation (clockwise from battery): $$+\mathcal{E} - IR_1 - IR_2 = 0$$ $$\mathcal{E} = I(R_1 + R_2)$$ $$I = \frac{\mathcal{E}}{R_1 + R_2}$$

Example 2: Multi-Source Loop

Circuit: Two batteries ($\mathcal{E}_1$, $\mathcal{E}_2$) and three resistors

KVL equation: $$\mathcal{E}_1 - IR_1 - IR_2 - \mathcal{E}_2 - IR_3 = 0$$

(Assuming both emfs are traversed (−) → (+) and current flows through all resistors)

Example 3: Complex Loop

From FAD1022 L12, Loop ABCFA: $$-2I_1 - 4I_1 - 3I_3 + 24 = 0$$ $$24 = 6I_1 + 3I_3$$

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Common Mistakes to Avoid

1. Wrong sign for emf: Remember (−) to (+) is +ℰ
2. Wrong sign for resistor: Go WITH current = drop (−IR)
3. Missing elements: Account for every component in the loop
4. Wrong loop count: Need enough independent loops for all unknowns

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Independent Loops

For a circuit with:

• $b$ branches
• $n$ nodes
• You need $b - (n - 1)$ independent KVL equations

Tip: Choose loops that include at least one element not in other loops.

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Related Concepts

• Kirchhoff's Current Law (KCL) — the companion law for junctions
• Kirchhoff's Rules — combined overview of both laws
• Multi-loop Circuits — applying both laws together
• FAD1022 L11 — Kirchhoff's Rules (Theory) — lecture source
• FAD1022 L12 — Kirchhoff's Rules (Applications) — worked examples

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Mermaid Diagram: KVL Around a Loop

flowchart LR
    A[Start at Point A<br/>V = 0] --> B[Pass through Battery<br/>V = +ℰ]
    B --> C[Pass through R₁<br/>V = ℰ - IR₁]
    C --> D[Pass through R₂<br/>V = ℰ - IR₁ - IR₂]
    D --> E[Return to A<br/>V = 0 ✓]
    
    style A fill:#90EE90
    style E fill:#90EE90

Key insight: After completing the loop, you must return to the starting potential.