Kirchhoff’s voltage law – KVL

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Kirchhoff’s voltage law also called as the Kirchhoff’s second law states that in any closed path of a network the algebraic sum of the voltages is equal to zero.

Kirchhoff’s laws are applicable only to lumped elements. This law may be applied to
constant DC source networks or networks driven a time variable sources like AC sources.

Example for Kirchhoff’s voltage law - KVL

Consider the following network of two voltage sources V₁, V₂ and two resistors R₁ and R₂ respectively. The voltage drops across the resistors R₁ and R₂ are given by V₃ and V₄ respectively.

Applying KVL to the circuit, it gives the following equation.

-V₁+V₃-V₂+V₄=0

⇒ V₁+V₂=V₃+V₄

Law of conservation of energy

Kirchhoff’s voltage law is based on law of conservation of energy. Consider the equation of the above example,

V₁+V₂=V₃+V₄

As voltage is the work done per unit charge i.e. \[v=\frac{dW}{dQ}\].

The equation can be represented as,

\[\frac{dW_{1}}{dQ}+\frac{dW_{2}}{dQ}=\frac{dW_{3}}{dQ}+\frac{dW_{4}}{dQ}\]

⇒ \[\frac{\frac{dW_{1}}{dt}}{\frac{dQ}{dt}}+\frac{\frac{dW_{2}}{dt}}{\frac{dQ}{dt}}=\frac{\frac{dW_{3}}{dt}}{\frac{dQ}{dt}}+\frac{\frac{dW_{4}}{dt}}{\frac{dQ}{dt}}\]

In a series circuit the current passing through all the elements is same. Hence dQ/dt is equal.

Therefore, W₁ + W₂ = W₃ + W₄

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