Kirchhoff's voltage law (KVL)

This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule.

The directed sum of the electrical potential differences(voltage) around any closed circuit must be zero.

Similarly to KCL, it can be stated as:

\sum_{k=1}^n V_k = 0

Here, n is the total number of voltages measured. The voltages may also be complex:

\sum_{k=1}^n \tilde{V}_k = 0

This law is based on the conservation of "energy given/taken by potential field" (not including energy taken by dissipation). Given a voltage potential, a charge which has completed a closed loop doesn't gain or lose energy as it has gone back to initial potential level.

This law holds true even when the resistance which cause dissipation of energy is present in a circuit. The validity of this law in this case can be understood if one realize that a charge in fact doesn't goes back to starting point, due to dissipation of energy. A charge will just terminate at the negative terminal, instead of positive terminal. This means all the energy given by the potential difference, has been fully consumed by resistance which in turn lose the energy as heat dissipation.

To summarize, Kirchhoff's voltage law has nothing to do with gain or loss of energy by electronic components (resistor, capacitor, etc). It is a law referring to the potential field generated by voltage sources. In this potential field, regardless of what electronic components are present, the gain or loss in "energy given by the potential field" must be zero when a charge completes a closed loop.