Kirchhoff’s Circuit Law – KCL and KVL

In this article, we will discuss Kirchhoff’s Circuit Laws – KCL (Kirchhoff’s Current Law) and KVL (Kirchhoff’s Voltage Law). Both KCL and KVL allow us to analyze and solve complex electric circuits. Using KCL and KVL, we can determine the values of electric currents and voltages in an electric circuit. Kirchhoff’s Current Law (KCL) is the basis of the nodal analysis technique whereas Kirchhoff’s Voltage Law (KVL) is the basis of the mesh analysis technique of circuit analysis. One major advantage of KCL and KVL is that they are equally valid for both DC circuits and AC circuits.

In the year 1847, German Physicist Gustav Robert Kirchhoff introduced two laws to describe the relationship between the currents and voltages in an electric circuit. Kirchhoff’s law along with Ohm’s law forms the basis of electric circuit theory.

From the basic theory of electric circuits, we have Ohm’s law to analyze electric circuits. But, Ohm’s law by itself is not enough to analyze all kinds of circuits. Although, when Ohm’s law is coupled with Kirchhoff’s laws, we get a set of powerful tools for analyzing a wide variety of electric circuits.

Kirchhoff’s first law, i.e. Kirchhoff’s Current Law is based on the law of conservation of electric charge, which means the algebraic sum of charges within a system remains constant.

Now, let us discuss Kirchhoff’s two laws (KCL and KVL) one by one-

Kirchhoff’s Current Law (KCL)

KCL is related to the currents at the junctions (a point in an electric circuit where three or more circuit elements meet) of an electric circuit and may be stated as under-

“The algebraic sum of currents meeting at a junction in an electric circuit is zero.”

Here, the algebraic sum is one in which the sign of the quantity is taken into account.

Explanation:

Consider five conductors carrying currents I₁, I₂, I₃, I₄, and I₅ meeting at a point O as shown in figure-1. Now, if we take the signs of electric currents entering point O as positive and the signs of electric currents leaving point O as negative. Then, according to KCL, we get,

`\I_1+I_2+(-I_3 )+I_4+(-I_5 )=0`

`\⟹I_1+I_2+I_4=I_3+I_5`

I.e.

Sum of incoming currents = Sum of outgoing currents

Hence, Kirchhoff’s Current Law (KCL) may also be stated as,

The sum of electric currents entering a node in an electric circuit is equal to the sum of electric currents leaving that node.

KCL is true for a node because the electric current is only the flow of electric charge, and it cannot be accumulated at any point in the circuit. This is in accordance with the law of conservation of electric charge.

Kirchhoff’s Voltage Law

Kirchhoff’s Voltage Law (KVL) is related to the EMFs and voltage drops in a closed circuit and may be stated as under-

In any closed electric circuit, the algebraic sum of all the electromotive forces and voltage drops in passive elements (ex. resistors) is equal to zero, i.e.

Algebraic sum of EMFs + Algebraic sum of the voltage drops = 0

Explanation:

Consider a closed electric circuit that is consisting of two resistors R₁ and R₂ in series connected across a source of emf E as shown in figure-2. Now, if we start from point A in this closed circuit and come back to this point A after going around the circuit. Then, there is no net change in the potential. This means the algebraic sum of EMFs of all the energy sources in the path plus the algebraic sum of the voltage drops in the passive elements must be zero.

In the given closed circuit, when we apply KVL, we get,

`\E+(-IR_1 )+(-IR_2 )=0`

`\⟹E=IR_1+IR_2`

The KVL is based on the law of conservation of energy. That means the net change in the energy of an electric charge after completing the closed path in an electric circuit is zero.

Sign Convention:

While applying KVL in a closed electric circuit, we deal with the algebraic sum of EMFs and voltage drops in the circuit. Therefore, it is important to use a proper sign of EMFs and voltage drops in the closed circuit. In actual practice, the following sign convention is followed:

A rise in potential across an element of the circuit is considered positive and a drop in the potential across an element of the circuit is considered negative.

Numerical Example (1) – Determine the electric current flowing through the 5 Ω resistor in the circuit shown in figure-3.

Solution – Applying Kirchhoff’s Current Law at node (a), we get,

Sum of incoming current = Sum of outgoing currents

`\15+10=I`

`\∴I=25" A"`

Hence, the current through 5 Ω resistor is equal to 25 A.

Numerical Example (2) – Determine the voltage drop across the 5 Ω resistor in the circuit shown in figure-4.

Solution – Applying Kirchhoff’s Voltage Law in the closed loop, we can write,

Sum of EMFs = Sum of Voltage Drops

`\14=4+V`

`\V=10" V"`

Hence, the voltage drop across the 5Ω resistor in the circuit is 10 Volts.

Conclusion

We may conclude the above discussion with the following points:

• Kirchhoff’s Current Law (KCL) states that in any electric circuit, the algebraic sum of currents entering a node is equal to the algebraic sum of leaving current at that node.
• KCL is based on the law of conservation of electric charge.
• KCL is also known junction rule of circuit analysis.
• Kirchhoff’s Voltage Law (KVL) states that in any closed electric circuit, the algebraic sum of all the EMFs equals the algebraic sum of all the voltage drops.
• KVL is based on the law of conservation of energy.
• KVL is also known as the loop rule or mesh rule of circuit analysis.

Related Articles

• Ohm's Law
• Nodal Analysis
• What is Electric Current?