Kirchhoff’s laws are fundamental principles in electrical circuit analysis. These laws provide a systematic approach to analyze complex circuits and find unknown voltages and currents.

## Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Voltage Law (KVL), also known as Kirchhoff’s second law, is a fundamental principle in electrical circuit analysis. It states that the algebraic sum of the voltage differences (voltages) around any closed loop or mesh in a network is always equal to zero. In other words, the total voltage rise in a closed loop is equal to the total voltage drop. This principle is based on the conservation of energy, as energy cannot be created or destroyed within a closed loop.

KVL can be mathematically expressed as:

ΣV = 0

Where ΣV is the sum of all voltage differences in the closed loop.

KVL is useful in analyzing electrical circuits, especially when determining unknown voltages, currents, or resistances. In combination with Kirchhoff’s Current Law (KCL), it forms the basis for various circuit analysis techniques, such as mesh analysis and nodal analysis, which are essential for understanding and designing complex electrical circuits.

To apply KVL in circuit analysis, follow these steps:

- Identify all closed loops or meshes in the circuit.
- Assign voltage polarities to each component based on the assumed direction of the current.
- Write KVL equations for each loop, summing the voltage differences around the loop and equating the sum to zero.
- Solve the resulting system of equations to determine the unknown voltages, currents, or resistances.

### Applications of KVL

- Circuit Analysis: KVL is used to analyze complex circuits, particularly those with multiple loops. By creating equations based on KVL for each loop, a system of linear equations can be formed and solved to determine unknown voltages or currents.
- Mesh Analysis: KVL is the foundation of mesh analysis, a method for analyzing circuits with multiple loops. By applying KVL to each loop (mesh), a set of linear equations can be derived and solved to find the mesh currents.
- Determining Voltage Drop: KVL can be used to calculate the voltage drop across individual components in a circuit, which is essential for designing and troubleshooting electronic and electrical systems.

## Example of Calculation

Let’s consider a simple DC circuit with one voltage source (V1), and three resistors (R1, R2, and R3) connected in a mesh configuration. The goal is to calculate the current flowing through each resistor using Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL).

Given values:

- V1 = 12 V (DC)
- R1 = 4 Ω
- R2 = 6 Ω
- R3 = 2 Ω

Step 1: Assign unknown currents to each resistor:

Let’s assume the unknown currents are I1, I2, and I3, for resistors R1, R2, and R3, respectively.

Step 2: Apply Kirchhoff’s Current Law (KCL) at the junctions:

At junction A (between R1 and R2), we have: I1 = I2 + I3

At junction B (between R2 and R3), we have: I3 = I2 + I1

Step 3: Apply Kirchhoff’s Voltage Law (KVL) around each loop:

Loop 1 (V1, R1, and R2): V1 – I1 * R1 – I2 * R2 = 0 12 – 4 * I1 – 6 * I2 = 0

Loop 2 (R2, R3, and I3):

- I2 * R2 – I3 * R3 = 0
- 6 * I2 – 2 * I3 = 0

Step 4: Solve the system of equations:

We have three equations with three unknowns (I1, I2, and I3):

- I1 = I2 + I3
- 12 – 4 * I1 – 6 * I2 = 0
- 6 * I2 – 2 * I3 = 0

Solving this system of equations, we find:

I1 ≈ 1.6 A I2 ≈ 0.8 A I3 ≈ 0.8 A

In conclusion, the current flowing through resistor R1 (I1) is approximately 1.6 A, and the current flowing through resistors R2 (I2) and R3 (I3) are approximately 0.8 A each.