Kirchhoff's Voltage Law (KVL)
Kirchhoff's voltage law: the sum of all voltages around any closed loop equals zero.
Fundamental to circuit analysis and mesh equations.
The Formula
Kirchhoff's voltage law states that the sum of all voltage rises and drops around any closed loop in a circuit equals zero.
This is based on the conservation of energy — a charge that travels around a loop returns to its starting potential.
Variables
| Symbol | Meaning |
|---|---|
| ΣV | Algebraic sum of all voltages in the loop (Volts, V) |
| V_source | Voltage rise from a power source (positive) |
| V_drop | Voltage drop across a component (negative) |
Sign Convention
- Voltage rises (e.g., going from − to + of a battery) are positive
- Voltage drops (e.g., going through a resistor in the direction of current) are negative
Example 1
A loop has a 12 V battery and two resistors. The first resistor drops 7 V. What is the voltage across the second resistor?
ΣV = 0
+12 V - 7 V - V₂ = 0
V₂ = 12 - 7
V₂ = 5 V
Example 2
A circuit loop has a 9 V battery and three resistors: R₁ = 100 Ω, R₂ = 150 Ω, R₃ = 50 Ω. Find the current and voltage drop across each resistor.
Total resistance: R_total = 100 + 150 + 50 = 300 Ω
Current: I = V / R = 9 / 300 = 0.03 A
V₁ = I × R₁ = 0.03 × 100 = 3 V
V₂ = I × R₂ = 0.03 × 150 = 4.5 V
V₃ = I × R₃ = 0.03 × 50 = 1.5 V
Check: 9 - 3 - 4.5 - 1.5 = 0 ✓
V₁ = 3 V, V₂ = 4.5 V, V₃ = 1.5 V (total = 9 V)
When to Use It
Use Kirchhoff's voltage law when you need to:
- Find unknown voltages across components in a loop
- Set up mesh equations for solving complex circuits
- Verify that voltage drops in a circuit are consistent
- Analyse circuits with multiple voltage sources
KVL applies to every closed loop in a circuit, no matter how many loops exist.
Combined with KCL, it provides all the equations needed to solve any linear circuit.