# Kirchhoff’s Laws

Gustav Robert Kirchhoff derived two fundamentals laws applicable to any electrical circuit. This law explains the algebraic sum of all branch voltages around a loop and all branch currents entering or leaving node.(References polarities and references directions are taken into account).

- Kirchhoff’s current Laws(KCL)
- Kirchhoff’s voltage Laws(KVL)

## 1.Kirchhoff’s current Laws(KCL)

Kirchhoff’s current Laws states that :”The algebraic sum of all branch currents leaving or entering a node is Zero at all instants of time”. KCL is* consequence of conservation of charge.* Charge which enters a node must leave that node because it may neither be created nor destroyed. Since the algebraic summation of charge must be Zero, the time derivative of this summation must be equal to Zero.

Consider the figure,

where i1,i2,i3,i4,i5 and i6 are the currents in the six branches meeting at node K of a circuit and having the references directions as indicated. Then for node K the KCL constraint among the six currents can be written as

where b is the number of branches meeting node K .

(If leaving current is taken as a references)

(If entering current is taken as a references)

Therefore,

In other words,

“The sum of the currents entering a node in a circuit through branches connected to that node at any instant of time is equal to the sum of the currents leaving the node at that instant through the remaining branches connected to the same node.”

## 2.Kirchhoff’s voltage Laws(KVL)

Kirchhoff’s voltage Laws(KVL) states that,” The algebraic sum of drop(or rise) in all branch voltage in a circuit, around a loop is Zero at all instants of time”. KVL is the ** consequence of conservation of energy **,voltage being the energy per unit charge, Since the algebraic summation of energy loss or gained must be Zero ,the algebraic summation of voltages also equal to Zero.

consider a figure,

where v1,v2,v3 and v4 are the branch voltages with references polarities assigned for the elements and a clockwise loop direction selected for the application of KVL.A positive sign is assigned for the voltage if the polarity marks occur in the order + to – and a negative sign for the opposite order i.e(- to +).

Now,

where b is the number of branches in a loop l.

In other words,

“The sum of the potential drops, at any instant of time, along the branches of a circuit when traversing a loop in a certain direction ,clockwise or counter-clockwise ,is equal to the sum of the potential rise in the remaining branches forming the loop.”

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