Kirchhoff's Law - Kirchhoff's Current Law & Kirchhoff's Voltage Law

By Mohit Uniyal|Updated : August 6th, 2022

Kirchhoff's Law is a powerful tool for analyzing electrical circuits. To analyze any electrical circuit, Ohm’s law is insufficient. Still, any electrical circuit can be analyzed with Kirchhoff's Law or the combination of Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL).

Kirchhoff’s law governs the electrical network's relationship between voltages and currents. It was introduced in 1847 by Gustav Robert Kirchhoff, a German physicist. The article elaborates on Kirchhoff's Law and equation, the validity of Kirchhoff's Law, and some examples.

Table of Content

What is Kirchhoff's Law?

In 1847, Gustav Robert Kirchhoff's introduced a pair of laws based on the law of conservation of charge and energy in an electrical circuit. The equivalent impedance of any complex network or circuit can easily be calculated using Kirchhoff's Law.

Define Kirchhoff's Law

The current or voltage of any circuit branch can also be calculated using Kirchhoff's Law. These laws are valid in AC and DC networks at low frequencies.

Kirchhoff's laws are classified into two types:

  • Kirchhoff's Current Law (KCL)
  • Kirchhoff's Voltage Law (KVL)

Kirchhoff's Current Law

Kirchhoff’s current law is also known as Kirchhoff's First law or Kirchhoff's Law of the junction, but the most used term is Kirchhoff's Current Law or KCL. KCL is based on the law of conservation of charge.

Define Kirchhoff's Current Law

Kirchhoff’s current law states that the algebraic sum of currents entering a node or a closed boundary equals zero.

If there are N number of branches connected to a node and it is the current of the nth branch, then mathematically, KCL states,

Nn=1i= 0

In this law, current entering a node is taken as positive, and current leaving a node is taken as negative or vice versa. Assume a set of currents ikt where k = 1, 2, 3 . . . . n flow into a node. Let iT(t) be the total current then,

Kirchhoff's Current Law

Current leaving=current entering

So, an alternate statement for KCL is that the sum of currents entering a node equals the sum of currents leaving the same node.

Application of Kirchhoff’s Current Law

KCL is used to combine the current sources present in parallel. The overall equivalent current is the algebraic sum of individual current present in parallel, as shown below:

Applying KCL at node a,

Application of Kirchhoff’s Current Law

Validity of Kirchhoff's Current Law

There exist some conditions where KCL is valid, and in some cases, it is not valid. Those conditions are:

  • KCL is independent of the variation in temperature in the circuit.
  • KCL is valid for linear, non-linear, bilateral, unilateral, passive, and active elements.
  • KCL is valid for lumped electrical networks only, not for distributed electrical networks. At high frequencies, the circuit is treated as distributed and not lumped, and the effect of parasitic resistance cannot be ignored, so KCL is invalid at high frequencies.
  • Kirchhoff’s law is not valid for time-varying magnetic fields.

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law is also known as Kirchhoff's Second law or KVL. KVL is based on the law of conservation of energy.

Define Kirchhoff's Voltage Law

Kirchhoff's Voltage Law states that the algebraic sum of voltages around a closed path or loop in a circuit equals zero. If there are M number of voltages in a loop and Vm is the mth voltage, then mathematically, KVL can be written as:

Mn=1v= 0

The sign of each voltage is taken as the polarity of the terminal encountered first as we travel around the loop. We can go around the loops in either clockwise or anti-clockwise directions. Consider the figure shown below:byjusexamprep

This shows that KVL is based on the law of conservation of energy.

Application of Kirchhoff's Voltage Law

KVL is used to combine the voltage sources present in series. The overall equivalent voltage is the algebraic sum of individual voltage present in series as shown below:Application of Kirchhoff's Voltage Law

Validity of Kirchhoff's Voltage Law

There exist some conditions where KVL is valid, and in some cases, it is not valid. Those conditions are:

  • KVL is independent of the variation in temperature in the circuit.
  • KVL is valid for linear, non-linear, bilateral, unilateral, passive, and active elements.
  • KVL is only valid for lumped electrical networks, not for distributed electrical networks.

Validity of Kirchhoff's Law

Kirchhoff's Laws are valid for lumped networks, not for distributed networks. There are some hypothetical cases where KCL and KVL are not satisfied.

  • Two unequal voltage sources are placed in parallel to each other.

If V1 ≠V2, then the KVL violates because KVL states that the sum of voltages in a closed loop must be zero.

Validity of Kirchhoff's Law

  • Two unequal current sources are placed in series to each other.

If I1 ≠I2, the KCL violates because the current entering at node a will not be equal to the current leaving at node a.

Validity of Kirchhoff's Law

Kirchhoff's Law Formula

Kirchhoff’s law comprises KVL and KCL. KVL states that the algebraic sum of voltages around a closed loop equals zero. KCL states that the algebraic sum of currents entering a node is equal to zero. Mathematically, KCL and KVL can be written as:

Kirchhoff's Law Formula

Kirchhoff's Law Examples

Example 1: Find current i0 and voltage v0 in the circuit shown below

Kirchhoff's Law Examples

Example 2: Find current i0 and voltage v0 in the circuit shown below

Kirchhoff's Law Examples 2

Example 3: Determine current i and voltage v0 in the circuit shown below

Kirchhoff's Law Examples 3

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FAQs on Kirchhoff's Law

  • Kirchhoff’s circuit laws are the heart of circuit analysis. With the help of these laws, along with ohm’s law, we have the essential tool to start analyzing any electrical circuits. Kirchhoff's Laws are broadly classified into KCL and KVL.

  • Kirchhoff’s law, along with ohm’s law is the heart to analyze any primary electrical circuit. Kirchhoff’s laws are used to explain the function of current and voltage in an electrical circuit. However, they can also be used to study complicated circuits that cannot be further simplified.

  • Kirchhoff’s law is primarily used in mesh and nodal analysis to find the unknown voltage and current across branches in an electric circuit. KCL and KVL are also useful to determine the equivalent resistance of some circuits where dependent sources are present.

  • Kirchhoff’s law formula is based on the law of energy and charge conservation. Formula based on law of energy conservation is known as KVL and formula based on law of charge conservation is known as KCL.

  • Kirchhoff’s first law is based on the law of conservation of charge also known as KCL. It states that the algebraic sum of currents entering a junction, node, or closed boundary equals zero. The alternate statement states that the sum of currents entering in a node is always equal to the sum of currents flowing out of the same node.

  • Kirchhoff’s second law is based on the law of energy conservation also known as KVL. It states that the algebraic sum of voltages in a closed loop always equals zero. The alternative statement asserts that the total of voltage drops across loop elements equals the sum of voltage rises from loop elements.

  • Kirchhoff's Law is not valid for distributed networks and for time-varying magnetic fields. At high frequency also, Kirchhoff's Law fails. 

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