# Kirchhoff’s law

Kirchhoff gave two laws for current and voltage. First law is called KCL and second law is called KVL.

**Kirchhoff’s current law (KCL):** This is the first law of Kirchhoff’s about current. How current divide in a circuit this law explains.

Gustav Robert Kirchhoff (Photo by wikipedia)

Kirchhoff’s current law is *∑ incoming current = ∑ Outgoing current*

At a point in a circuit the value of incoming current and outgoing current is equal.

Let consider the figure i1 and i3 are incoming currents in O point i2 , i4 , i5 , i6 are the outgoing currents. The value of incoming currents two paths are same as the value of outgoing currents four paths. Here path is not fact, fact is that incoming and outgoing current value will be same.

According to this law mathematically,

i1 + i3 = i2 + i4 + i5 + i6

i1 + i3 + (- i2 ) + (- i4 ) +(- i5 ) +(- i6) = 0

In another word KCL states that* “The algebraic sum of currents in a network of conductors meeting at a point is zero.”*

Consider above circuit according to Kirchhoff’s current law at A point. I is incoming current and there is two outgoing currents are I1 , I2 . Mathematically, I = I1 + I2

Practical example of KCL is our household. In our house incoming current enters through only one way and it divides many paths by our electronics equipments like TV, fridge, bulb, fan etc. Here incoming current path is only one but outgoing is many.

**Kirchhoff’s voltage law (KVL):** KVL is the second law of voltage by Kirchhoff. Before approaching KVL we have must knowledge about circuit element, node, branch and loop. Then KVL will be easy to understand.

The elements which are used in circuit is called circuit element. Resistor, capacitor, cell etc are the circuit element.

**Node:** A circuit element has two terminals. Any one terminal of element is called node. Many terminals of circuit element can be connected in one node. See above picture there is three nodes.

**Branch:** There are two terminals in one circuit element. The two terminals of one circuit element is called branch. So one circuit element has one branch. In above picture there are five branches are marked in circle shape.

**Loop:** If any circuit has two or more sources at a time and has many circuit elements. In that case we divide the total circuit into many small portions. The small portion is known as loop. A loop is considered from a node supplying current over some branches and must end the loop where first node considered. In above image there are three loops can be considered which are shown in red circle.

**Kirchhoff’s voltage law (KVL):** Kirchhoff’s second law is for voltage which is also called KVL.

According to this law, *∑ Supply voltage = ∑ Drop voltage*

Supply voltage in a circuit is equal to drop voltage of the circuit. To understand easily let consider an example: A boy named Emon went to School from home taking $100. He expensed $15 dollar taxi bill, bought an ice cream $10 finally came back home $15 by taxi bill. Totally he expensed $40 and remaining balance was $60. Here supply is $100 and drop is $60.

KVL is something like that supply voltage to a circuit and drop voltage is always equal for the same circuit. Here drop voltage means drop and remaining voltage.

Consider the above circuit. It has two supply source and three resistors. Now we calculate this circuit to verify Kirchhoff’s voltage law. Resistors positive terminal should be connected with source positive terminal. There are two sources so we can make it two separate loops. One thing we should remind that we have to end that node counting from where we start. Another thing is minus or plus sign counting. Some use clock wise anti clock wise system. Here I am using another method which is very easy than clock counting. We consider going through minus to plus sign is positive and plus to minus sign is negative for circuit element. You can take reverse it but result will be same.

– + = +

+ – = –

For bafeb loop,

*+ E1 – V1 – V2 = 0*

* or E1 – I1R1 – I2R2 = 0*

* or E1 = I1R1 + I2R2 *

* E1 = ∑ IR*

So that we can say supply voltage and drop voltage is equal for first loop bafeb.

For another second loop bedcb,

*+ V2 + V3 – E2 = 0*

* or + I2R2 + I3R3 – E2 = 0*

* or E2 = I2R2 + I3R3*

* E2 = ∑ IR*

From this equation we can say supply voltage and drop voltage is same for second loop bedcb.

Hence for two loops supply voltage and drop voltage is equal.

*∑ E = ∑ IR*

Alternatively,

Kirchoff’s law (KVL) states that *“Algebraic sum of all voltages for a close path is zero.”*

Mathematically KVL is expressed that

Where M is the number of voltages in the loop.

Let, we start from voltage source V1 and consider from negative to positive is negative sign and positive to negative is positive sign.

Now applying KVL we get,

-V_{1} + V_{2 }+ V_{3 }+ V_{4 }– V_{5 }+ V_{6} = 0

Or V_{2 }+ V_{3 }+ V_{4 }+ V_{6} = V_{1 }+ V_{5}

It means sum of voltage drops = sum of voltage rises.

This is an alternative equation of KVL law.