In 1845, a German physicist, Gustav Kirchhoff developed a pair or set of rules or laws which deals with the conservation of current and energy within electrical circuits. These two rules are commonly known as Kirchhoff’s Circuit Laws with one of Kirchhoff’s law dealing with the current flowing around a closed circuit known as Kirchhoff’s Current Law (KCL), while the other law deals with the voltage sources present in a closed circuit known as Kirchhoff’s Voltage Law (KVL). The Kirchhoff’s Laws are very useful in solving electrical networks which may not be easily solved by Ohm’s Law.

Kirchhoff’s First Law – The Current Law, (KCL)

Kirchhoff’s current law or KCL, states that the:

“Total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node.”

In other words, the algebraic sum of all the currents entering and leaving a node must be equal to zero.

I(exiting) + I(entering) = 0. 

This idea by Kirchhoff is commonly known as the conservation of charge.

Kirchhoff’s Current Law

Here, the three currents entering the node, I1, I2, I3 are all positive in value, and the two currents leaving the node, I4 and I5 are negative in value. Then this means we can also rewrite the equation as;

I1 + I2 + I3 – I4 – I5 = 0

The term node in an electrical circuit generally refers to a connection or junction of two or more current-carrying paths or elements such as cables and components. Also, for current to flow either in or out of a node a closed-circuit path must exist. We can use Kirchhoff’s current law when analyzing parallel circuits.

Kirchhoff’s Second Law – The Voltage Law, (KVL)

Kirchhoff’s voltage law or KVL, states that

 “In any closed-loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop” which is also equal to zero. 

In other words, the algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchhoff is known as the conservation of energy.

Or

The sum of all voltages around any closed path (loop) in a circuit equals zero.  

      ∑voltage rise =∑voltage drop  

Kirchhoff’s Voltage Law  Explanation

Starting at any point in the loop, continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use Kirchhoff's voltage law when analyzing series circuits. When analyzing either DC circuits or AC circuits using Kirchhoff’s Circuit laws, several definitions and terminologies are used to describe the parts of the circuit being analyzed such as nodes, paths, branches, loops, and meshes.These terms are used frequently in circuit analysis, so it is important to understand them.

Common DC Circuit Theory Terms:

• Circuit – A circuit is a closed-loop conducting path in which an electrical current flows..
• Path – A single line of connecting elements or sources.
• Node – A node is a junction, connection, or terminal within a circuit where two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot. 
• Branch – A branch is a single or group of components such as resistors or a source which are connected between two nodes.
• Loop – A loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once.
• Mesh – A mesh is a single open loop that does not have a closed path. There are no components inside a mesh.

Application of Kirchhoff’s Circuit Laws

The applications include:

• They can be used to analyze any electrical circuit.
• Computation of current and voltage of complex circuits.
• Calculation of unknown currents and voltages is easy.
• Simplification and analysis of complex closed-loop circuits becomes manageable.

Kirchhoff’s Circuit Law Example No1

Find the current flowing in the 40Ω Resistor, R3

The circuit has 3 branches, 2 nodes (A and B), and 2 independent loops.

Using Kirchhoff’s Current Law, KCL the equations are given as:

At node A :    I1 + I2 = I3  

At node B :    I3 = I1 + I2

Using Kirchhoff’s Voltage Law, KVL the equations are given as:

Loop 1 is given as :    10 = R1 I1 + R3 I3 = 10I1 + 40I3

Loop 2 is given as :    20 = R2 I2 + R3 I3 = 20I2 + 40I3

Loop 3 is given as :    10 – 20 = 10I1 – 20I2

As I3 is the sum of I1 + I2 we can rewrite the equations as;  

Eq. No 1 :    10 = 10I1 + 40(I1 + I2)  =  50I1 + 40I2

Eq. No 2 :    20 = 20I2 + 40(I1 + I2)  =  40I1 + 60I2  

We now have two “Simultaneous equations” that can be reduced to give us the values of I1 and I2 

Substitution of I1 in terms of I2 gives us the value of I1 as -0.143 Amps

Substitution of I2 in terms of I1 gives us the value of I2 as +0.429 Amps

As :    I3 = I1 + I2

The current flowing in resistor R3 is given as : -0.143 + 0.429 = 0.286 Amps

and the voltage across the resistor R3 is given as :  0.286 x 40 = 11.44 volts

The negative sign for I1 means that the direction of the current flow initially chosen was wrong, but still valid. The 20v battery is charging the 10v battery.