6: Kirchhoff's Voltage Law (KVL)

Kirchhoff's Voltage Law (KVL)

Fundamental Principle:
Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around any closed loop in a circuit is zero. This principle arises from the conservation of energy: energy gained by charge carriers (e.g., from voltage sources) must equal energy lost (e.g., across resistors).

Mathematical Statement:
For a closed loop with n voltage drops/rises:
$\sum_{k=1}^{n} V_k = 0$

Sign Convention:

• Voltage Drop (+): When traversing the loop, if you encounter a component from + to – (e.g., moving through a resistor in the direction of current flow).
• Voltage Rise (–): When traversing from – to + (e.g., passing through a battery’s negative to positive terminal).

Application Steps:

1. Define Loop Direction: Choose clockwise or counterclockwise traversal.
2. Assign Polarities: Mark component polarities (current direction determines resistor voltage polarity: + where current enters).
3. Sum Voltages Algebraically: Add voltage rises and drops with signs based on traversal direction.

Example:
Consider a loop with a 10V battery and two resistors (R₁ = 2Ω, R₂ = 3Ω). Current I flows clockwise.

• Battery (10V): Traversal from – to + → –10V (rise).
• R₁ (2Ω): Voltage drop = $I \times 2\Omega$ → +2I (drop).
• R₂ (3Ω): Voltage drop = $I \times 3\Omega$ → +3I (drop).
  KVL Equation:
  $-10 + 2I + 3I = 0 \implies 5I = 10 \implies I = 2 \text{ A}$

Key Implications:

1. Loop Independence: KVL applies independently to all closed loops (meshes) in a circuit.
2. Multiple Sources: Voltage rises (e.g., batteries) subtract from drops (e.g., resistors).
3. Conflict Handling: If a loop contains opposing voltage sources, their net value dictates current direction.

Common Exam Focus:

• Writing KVL equations for multi-loop circuits (e.g., using mesh analysis).
• Verifying power balance (e.g., power supplied by sources equals power absorbed by resistors).
• Analyzing circuits with dependent sources (treat as variables in KVL).

Critical Mistake to Avoid:
Inconsistent sign assignment during loop traversal. Always:

• Subtract voltage sources when exiting the + terminal.
• Add drops across passive components when moving with current flow.

Practical Tip:
For complex loops, "walk" the loop path mentally, noting each component’s contribution as a rise (+) or drop (–) relative to your direction.