How to Apply KVL (Kirchhoff Voltage Law) in Circuits with Examples

Learn how to apply Kirchhoff’s voltage law in circuits with solved examples. Kirchhoff’s voltage law, KVL, or Kirchhoff’s second law, deals with the conservation of energy in a closed-loop electric circuit. KVL is the fundamental law for electrical circuit analysis.

What does Kirchhoff’s Voltage Law state?

The Kirchhoff Voltage Law KVL states that the algebraic sum of the voltage produced and the voltages dropped in a closed loop (a closed path) of an electric circuit is always equal to zero. 

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Mathematically, 

In other words, the energy is conserved across a closed loop of an electric circuit.

Note that the algebraic sum means taking into account the polarities and signs of voltage sources and voltage drops. When applying KVL to any circuit, all the signs (+ -) or (- +) must be taken into account correctly, the calculation will be wrong.

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Gustav Kirchhoff, a German physicist, presented two laws: Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL). Ohm’s law is very basic and may not be sufficient to analyze a complex circuit. Kirchhoff’s Voltage Law KVL provides the basis for Mesh Current Analysis.

How to Apply KVL: The 3-Step Rule

Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of all voltages around a closed circuit loop is zero.

Follow these three steps to apply KVL:

1. Choose a closed loop: Select a complete path in the circuit and decide the direction of traversal (clockwise or anticlockwise).
2. Assign voltage polarities: Mark voltage rises as positive (+) and voltage drops as negative (−) according to your chosen direction.
3. Write and solve the equation: Add all voltage rises and drops around the loop and set their algebraic sum equal to zero.

KVL Formula:

This means that the total voltage supplied in a closed loop equals the total voltage dropped across its components.

How to Write a KVL Equation in 3 Easy Steps

Writing a Kirchhoff’s Voltage Law (KVL) equation becomes easier when you follow a consistent method. The main idea is to calculate all voltage rises and drops around a closed loop and make their algebraic sum equal to zero.

Let’s understand the process step by step.

Step 1: Identify the Closed Loop and Current Direction

First, identify a complete closed loop in the electrical circuit. Choose a direction to travel around the loop, either clockwise or anticlockwise.

If the circuit contains resistors, assign a current direction. You can assume the current direction if it is not already known. If your calculated current is negative, the actual direction is opposite to your assumption.

Step 2: Assign Voltage Polarities

Identify the positive and negative terminals of each voltage source and resistor.

Use these sign conventions while moving around the selected loop:

• Moving from negative (−) to positive (+) across a voltage source represents a voltage rise, so write it as positive.
• Moving from positive (+) to negative (−) across a voltage source represents a voltage drop, so write it as negative.
• Moving through a resistor in the direction of conventional current represents a voltage drop, written as −IR.
• Moving through a resistor opposite to the current direction represents a voltage rise, written as +IR.

Step 3: Write and Solve the KVL Equation

Add all voltage rises and drops around the closed loop and set their algebraic sum equal to zero.

The general KVL equation is:

For a simple DC circuit containing one voltage source and multiple resistors:

Rearrange the equation to calculate the unknown current or voltage.

Kirchhoff’s Voltage Law Solved Example

Suppose a circuit with two parallel paths (loops) and a single voltage source (DC), as shown in the diagram below. Find the current and voltage of each element of the circuit for the following given circuit parameters using Kirchhoff’s voltage law.

R₁ = 5Ω
R₂ = 10Ω
R₃ = 5Ω

R₄ = 10Ω
V = 20Volts

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Example Solution:

Assign Loop Names:

There are two loops (closed paths) in the circuit: loop 1 with two resistors and a single voltage source, whereas in loop 2, there is no voltage source, only three resistors.

Assign Sign to Circuit Elements:

Now, assign a positive sign to the resistor terminal that is closer to the voltage source’s positive terminal, and a negative sign to those that are closer to the negative terminal. OR you may assign a (+ -) sign to each component terminal by going through the loop. But make sure that the sign of voltage rise (- +) is exactly opposite to the sign of voltage drop (+ -). Both will give you the same results, even the same equations.

Assign Current Names:

Assign a name to the current of each loop, and as discussed in KCL, write equations for the current at each node. For the above circuit, KCL equations will be:

Write Loop Equations:

The next step is to write equations for each loop. Based on the sign and current name assigned, as shown below.

Notice the negative sign in the second equation; it is because it is opposite in the direction of the loop arrow, i.e., for R2 and R3 it is (+ -), but for R₄ it is (- +).

Now, put in the value of I₁, which will give you two simultaneous equations with unknown variables I₂ and I₃. The value of these currents can easily be obtained using Cramer’s rule for the matrix.

Let’s put the values of the resistors and the voltage source, and see what happens

Now multiply equation (1) by -3 and add it to equation (2)

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Solve The Equations:

Adding the above two equations will give us equation (3)

OR

Now, put the current I₂ into equation (2) to get the current I₃

OR

By applying KCL to node b, the following equation can be obtained

Now, putting the values of the current will give us the value of I₁

Now, it is easy to find any parameter of the circuit; just use Ohm’s law, as shown below

And

Does Kirchhoff’s voltage law apply to parallel circuits?

Yes, KVL applies to the parallel circuit. First, identify the loops and apply the KVL. The circuit in the above example is a parallel circuit.

Kirchhoff’s Voltage Law Limitation:

The KVL has some practical limitations, but it is a very useful tool for circuit analysis.

1. In a high-frequency circuit, the fluctuating magnetic field can link the circuit, which contradicts the assumption of KVL.  

Short Answer Questions:

Can Kirchhoff’s voltage law be applied to an incomplete loop?

Kirchhoff’s second law requires that the loop is closed. The open loop will not allow the current to flow, and there will be no voltage drop across the circuit elements. All the voltage in the loop will appear across the open portion.

What is a parallel circuit?

A parallel circuit is an electrical arrangement where components are connected side by side, allowing multiple paths for current flow. Voltage across components remains constant, while current divides among them.

Does Kirchhoff’s voltage law apply to parallel circuits?

Yes, Kirchhoff’s Voltage Law applies to parallel circuits. It states that the sum of the voltages across all components in any closed loop within the circuit is zero.