Kirchhoff’s Voltage Law: Definition, Formula, Sign Convention and Example

What Is Kirchhoff’s Voltage Law?

Kirchhoff’s Voltage Law is also called Kirchhoff’s loop rule. It is based on the conservation of energy. When a charge moves around a complete circuit loop and returns to its starting point, its total change in electrical potential must be zero.

A voltage source, such as a battery, gives energy to electric charges. Resistors, lamps, motors and other components remove or transform that energy. The energy gained from the source must equal the energy transferred to the components.

Kirchhoff’s Voltage Law Formula

The mathematical form of KVL is:

ΣV = 0

This means that the algebraic sum of all voltage changes around a closed loop equals zero.

For a circuit containing one voltage source and several resistors, the equation may be written as:

V_source − V₁ − V₂ − V₃ = 0

Using Ohm’s law, V = IR, this becomes:

V_source − IR₁ − IR₂ − IR₃ = 0

Therefore:

V_source = IR₁ + IR₂ + IR₃

Why Does KVL Work?

Voltage represents electrical potential energy per unit charge. As a charge travels through a circuit, it may gain energy from a battery and lose energy in resistors or other components.

After traveling around a complete loop, the charge returns to its original point. It cannot have a net gain or loss of energy from one complete trip. Therefore, the total voltage change must be zero. This is why KVL is an application of the law of conservation of energy. ([openstax.org](https://openstax.org/books/university-physics-volume-2/pages/10-3-kirchhoffs-rules?utm_source=openai))

Voltage Rise and Voltage Drop

To apply KVL, identify whether each circuit element causes a voltage rise or a voltage drop in the direction you choose to travel.

• A voltage source is usually treated as a voltage rise when moving from its negative terminal to its positive terminal.
• A voltage source is usually treated as a voltage drop when moving from its positive terminal to its negative terminal.
• Across a resistor, moving in the direction of the assumed current gives a voltage drop of IR.
• Across a resistor, moving opposite to the assumed current gives a voltage rise of IR.

The direction chosen for traveling around the loop is arbitrary. You may choose clockwise or counterclockwise, but you must use the same direction consistently throughout the calculation.

How to Apply Kirchhoff’s Voltage Law

1. Identify a closed loop in the circuit.
2. Choose a direction around the loop, either clockwise or counterclockwise.
3. Assume a current direction if the current is unknown.
4. Assign signs to every voltage rise and voltage drop.
5. Write the KVL equation using ΣV = 0.
6. Use Ohm’s law, V = IR, when the voltage across a resistor is not known.
7. Solve the equation for the unknown current, voltage or resistance.
8. Check the result by confirming that the voltage rises and drops balance.

Simple Numerical Example

Consider a single-loop circuit with:

• A 12 V battery
• A 2 Ω resistor
• A 4 Ω resistor
• Both resistors connected in series

Because the resistors are in series, the same current flows through both resistors. Let the current be I.

Step 1: Write the KVL equation

Traveling around the loop in the direction of the current:

12 − I(2) − I(4) = 0

Step 2: Combine the resistance terms

12 − 6I = 0

Step 3: Solve for current

6I = 12

I = 2 A

Step 4: Find the voltage across each resistor

For the 2 Ω resistor:

V₁ = IR₁ = 2 × 2 = 4 V

For the 4 Ω resistor:

V₂ = IR₂ = 2 × 4 = 8 V

Check using KVL

12 V − 4 V − 8 V = 0

The result satisfies Kirchhoff’s Voltage Law. The 12 V supplied by the battery equals the total voltage drop across the resistors.

KVL in Circuits with Multiple Loops

In a complex circuit, there may be two or more closed loops. KVL can be applied separately to each independent loop.

When a circuit has multiple loops:

• Assign a current to each branch or loop as needed.
• Choose a direction for each loop.
• Write one KVL equation for each independent loop.
• Use additional circuit relationships, such as current rules at junctions, when necessary.
• Solve the resulting equations together.

Kirchhoff’s rules are especially useful when a circuit cannot be simplified easily using only series and parallel resistor rules. ([openstax.org](https://openstax.org/books/university-physics-volume-2/pages/10-3-kirchhoffs-rules?utm_source=openai))

Common Mistakes When Using KVL

• Forgetting that KVL applies to a closed loop: The path must return to its starting point.
• Mixing sign conventions: Choose a clear direction and apply the same voltage-rise and voltage-drop rules throughout.
• Using the wrong resistor voltage: The voltage across a resistor is IR, not just R.
• Assuming a negative current is an error: A negative answer usually means the actual current flows opposite to the assumed direction.
• Adding all voltages as positive: Voltage rises and voltage drops must be given appropriate signs.
• Confusing KVL with Kirchhoff’s Current Law: KVL deals with voltage around a loop, while the current law deals with current entering and leaving a junction.

Important Points to Remember

• Kirchhoff’s Voltage Law is based on conservation of energy.
• It applies to every closed circuit loop.
• The algebraic sum of voltage changes around a loop is zero.
• The total voltage rise equals the total voltage drop.
• For a resistor, the voltage magnitude is found using V = IR.
• The direction chosen for loop analysis is arbitrary, but it must remain consistent.
• A negative answer can indicate that the assumed current direction is opposite to the actual direction.