Kirchhoff's Laws - KCL and KVL

Gustav Robert Kirchhoff, a German physicist, gave two laws to solve complex circuits where ohm's law is not sufficient.

Kirchhoff's current law

is also called kirchhoff's first law or KCL or kirchhoff's junction rule.

It is stated as " The algebraic sum of all the currents meeting at a junction in an electrical circuit is zero " .

The term algebraic sum means that signs of the currents must be taken into account.

While applying KCL , you can take sign of current coming towards the junction as either +ve or –ve , depending on your choice. But once you have chosen the sign of incoming current, the sign of outgoing current must be taken opposite of it .

Lets understand by the example .

Here lets take the currents coming towards to be +ve , then , currents leaving the junction must be taken as –vely signed.

i.e +I₁, +I₄, -I₃and -I₂

According to kirchhoff's current law , the sum of all these currents with their respective signs must equal zero i.e

I₁+ I₄- I₃-I₂ = 0

or I₁+ I₄= I₃+I₂ ...(1)

i.e sum of incoming currents = sum of outgoing currents

Kirchhoff's Current Law is based on the Law of Conservation of Charge

Lets multiply (1) by time t , we get

I₁t + I₄t= I₃t+ I₂t

But, charge (Q) = current × time

∴ Q₁+ Q₄= Q₃+Q₂

Amount of charge coming towards the junction = Amount of charge leaving the junction

Showing that charge cannot accumulate in any part of an electric circuit and is conserved.

Kirchhoff's Voltage law

is also called kirchhoff's second law or KVL or kirchhoff's loop rule.

It is stated as " In any closed electrical circuit or mesh , the algebraic sum of all the emfs and voltage drops in resistors is equal to zero " .

While applying KVL to a closed circuit , the rise in potential should be considered positive and fall in potential should be considered negative.

Therefore , potential change while moving from –ve to +ve terminal of a cell or battery is taken positive and the potential change in a resistor while traversing it against the direction of current flow is taken positive.

The reverse of the rule is also true .

Lets apply the above rules to write down KVL equation for below circuit.

Let the loop be traversed in the direction shown. While moving in this direction , resistors are traversed along current direction ∴ IR₁ and IR₂ should be taken negative and battery voltage E as +ve as it is traversed from –ve to +ve terminal.

So , the loop equation is

- IR₁- IR₂ + E = 0

Kirchhoff's Voltage law is based on the Law of Conservation of Energy

Lets multiply above equation by current I and time t , we get

- I²R₁t– I²R₂t+ IEt= 0

energy supplied by E = energy consumed by R₁ + energy consumed by R₂

So , KVL follows the law of conservation of energy.

Important Points :-

(1) In case of ac circuits, the KCL and KVL must use vector sum instead of algebraic sum.
(2) Uses :-Kirchhoff's laws are applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear.
(3) Limitations :- KVL and KCL cannot be used for distributed network models.