Kirchhoff’s Laws

07/05/2026

The online simulations of Kirchhoff’s Laws on this page allow you to analyze circuits without needing to simplify them beforehand. Using virtual setups with nodes, loops, sources, resistors, bulbs, ammeters, and voltmeters, you can observe how current and voltage are distributed in each part of the circuit and how to apply the two fundamental laws—the current law and the voltage law—to describe its operation. These activities help you understand that any circuit, no matter how complex, can be studied using equations based on the conservation of charge and energy.

This Thematic Unit is part of our Circuits collection

Electrical Node

Point in a circuit where three or more conductors connect, essential for identifying parallel branches.

Kirchhoff’s First Law (KCL)

States that the sum of currents entering a node is equal to the sum of currents leaving it.

Kirchhoff’s Laws

Set of two laws based on the conservation of charge and energy, fundamental for the analysis of complex circuits.

Kirchhoff’s Second Law (KVL)

States that the algebraic sum of voltage drops and electromotive forces in a closed loop is equal to zero.

Mesh (Loop)

Any closed path in a circuit that does not contain other closed paths within it.

Mesh Equation

Mathematical expression resulting from applying the voltage law to a closed path to find unknown currents.

What are Kirchhoff’s Laws?

Kirchhoff’s Laws are two fundamental principles that allow you to analyze any electrical circuit, no matter how complex, without needing to simplify it first. They are based on two essential ideas from physics: the conservation of electric charge and the conservation of energy. The first applies to nodes and describes how currents are distributed when several conductors meet; the second applies to loops and explains how voltages are distributed in a closed path. Thanks to these laws, it is possible to set up equations that describe the behavior of the entire circuit and solve them to obtain unknown currents and voltages.

Kirchhoff’s Current Law (KCL)

Kirchhoff’s Current Law states that at any node in a circuit, the sum of the currents entering is equal to the sum of the currents leaving. This occurs because electric charge cannot accumulate at a point: all charge that enters must continue its path. With this law, we can analyze how current is distributed when a conductor splits into several paths, setting up equations that relate the current in each branch. It is a fundamental tool for studying circuits with multiple branches without needing to simplify them.

Kirchhoff’s Voltage Law (KVL)

Kirchhoff’s Voltage Law states that in any closed loop of a circuit, the algebraic sum of all voltages is zero. This means that the energy supplied by the source must be distributed exactly among all the elements in the loop, so that the voltage rises and drops balance out. Applying this law allows you to relate the voltage drops across resistors, bulbs, or other components to the voltage supplied by the source, and it is essential for analyzing circuits with multiple loops or possible paths. With it, you can set up equations that describe how electrical energy is distributed along each path of the circuit.

How to apply Kirchhoff’s Laws in a circuit

To apply Kirchhoff’s Laws in a circuit, you first need to identify its nodes and loops, since each law is used in a different place. The Current Law is applied at nodes to relate the currents distributed among various branches, while the Voltage Law is used in loops to connect the voltage drops across each component to the voltage supplied by the source. The usual procedure is to assign directions to the currents (even if they later turn out to be negative), write the corresponding equations for each node and each loop, and solve the resulting system. This method allows you to analyze complex circuits without reducing them, obtaining all the internal currents and voltages from conservation principles.

Applications of Kirchhoff’s Laws

Kirchhoff’s Laws allow you to solve circuits that cannot be easily simplified using equivalent resistances, such as those with multiple loops, many nodes, or irregularly distributed components. With them, it is possible to determine unknown currents and voltages in any part of the circuit, analyze how different branches behave when they share common nodes, and check if a setup meets the conservation of charge and energy conditions. They are also used to experimentally verify instrument readings, design more complex circuits, and understand how real electrical networks work, where simplifications are not always possible.

Electrical Node

Point in a circuit where three or more conductors connect, essential for identifying parallel branches.

Kirchhoff’s First Law (KCL)

States that the sum of currents entering a node is equal to the sum of currents leaving it.

Kirchhoff’s Laws

Set of two laws based on the conservation of charge and energy, fundamental for the analysis of complex circuits.

Kirchhoff’s Second Law (KVL)

States that the algebraic sum of voltage drops and electromotive forces in a closed loop is equal to zero.

Mesh (Loop)

Any closed path in a circuit that does not contain other closed paths within it.

Mesh Equation

Mathematical expression resulting from applying the voltage law to a closed path to find unknown currents.

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Simulations of Kirchhoff's Laws

Kirchhoff’s Current Law (KCL)

In this simulation, we use for the first time the schematic representation of circuits, which shows the components using electrical symbols instead of drawings. This format is more suitable for analyzing nodes, branches, and loops, and will allow us to apply Kirchhoff’s Laws clearly and rigorously. In this simulation, a node is represented from which three branches with different resistances depart, all powered by the same source. An ammeter placed before the node measures the total current entering, while three other ammeters record the current flowing through each branch. When you change the values of the resistances or the source voltage, you can observe how the distribution of currents changes, but it always holds that the incoming current is equal to the sum of the outgoing currents. This direct visualization helps to intuitively understand Kirchhoff’s Current Law and shows that electric charge does not accumulate at the node, but is distributed among the various available paths. Check that the sum of currents measured at the node matches the prediction of Kirchhoff’s Current Law (KCL).

Kirchhoff’s Voltage Law (KVL)

In this simulation, a simple loop is built formed by a voltage source and several resistors connected in series. Using two voltmeters, you can measure the voltage drop across each resistor and check that their sum matches the total voltage supplied by the source. By changing the values of the resistors or the source voltage, you can see how the individual drops change, but it always holds that the algebraic sum of all voltages in the closed loop is zero. This experience allows you to directly visualize Kirchhoff’s Voltage Law and understand that the energy delivered by the source is distributed exactly among all elements of the loop. Check that the sum of the measured voltage drops in the loop matches the voltage from the source, as established by Kirchhoff’s Voltage Law (KVL).

Double loop with shared node

In this simulation, a circuit is presented consisting of two loops that share a central resistor, which prevents it from being simplified using equivalent resistances. Each loop includes its own path with resistors, and both are connected at a common node. Ammeters are placed in each branch to measure the currents in each loop, and voltmeters at strategic points to record the voltage drops. By changing the values of the resistors or the source voltage, you can see how the currents in each loop change and how the shared resistor influences both loops. This setup allows you to simultaneously apply the Current Law (KCL) at the node and the Voltage Law (KVL) in each loop, showing how both laws combine to analyze circuits that can no longer be solved by simplification methods. Check that the voltages and currents measured in each loop match the values obtained theoretically using Kirchhoff’s Laws.

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Michael Faraday

1791

–

1867

Michael Faraday discovered electromagnetic induction, conducted pioneering experiments in optics (Faraday effect), and established fundamental principles of electrochemistry.

“Nothing is too wonderful to be true”

André-Marie Ampère

1775

–

1836

André-Marie Ampère formulated the theory of electromagnetism, establishing the mathematical foundations linking electricity and magnetism

“Science is the explanation of the complex by the simple”

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Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Michael Faraday

1791

–

1867

Michael Faraday discovered electromagnetic induction, conducted pioneering experiments in optics (Faraday effect), and established fundamental principles of electrochemistry.

“Nothing is too wonderful to be true”

André-Marie Ampère

1775

–

1836

André-Marie Ampère formulated the theory of electromagnetism, establishing the mathematical foundations linking electricity and magnetism

“Science is the explanation of the complex by the simple”

Become a giant

Your path to becoming a giant of knowledge begins with these top free courses

Free mode

Principles of Modeling, Simulations, and Control for Electric Energy Systems

Free mode

Principles of Electric Circuits | 电路原理

Free mode

Electrotechnique I

Free mode

Electromagnetic Compatibility Essentials

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Test your knowledge

What are Kirchhoff’s Laws?.

Kirchhoff’s Laws are two fundamental principles that allow any electrical circuit to be analyzed without needing to simplify it first. They are based on two essential ideas in physics: the conservation of electric charge and the conservation of energy. The first applies to nodes and describes how currents are distributed when several conductors meet. The second applies to loops and explains how voltages are shared along a closed path. Thanks to these laws, it is possible to set up relationships that describe the behavior of the entire circuit and use them to determine unknown currents and voltages in any part of the system. This makes them an essential tool for studying real electrical circuits.

What does Kirchhoff’s Current Law explain?

Kirchhoff’s Current Law states that at any node, the sum of the currents entering is equal to the sum of the currents leaving. This happens because electric charge cannot accumulate at a single point: all the charge that enters must continue its path. This law makes it possible to analyze how current is divided when a conductor splits into several branches, relating the currents in each path without needing to simplify the circuit. It is especially useful in circuits with multiple junctions, where current is distributed differently depending on the resistance of each branch. With this law, equations can be written that describe the behavior of the entire circuit.

What does Kirchhoff’s Voltage Law describe?

Kirchhoff’s Voltage Law states that in any closed path, the sum of all rises and drops in voltage must balance out. This means that the energy supplied by the source is shared exactly among the components of the circuit, so no energy is gained or lost along the way. Applying this law allows the voltage drops across resistors, bulbs or other devices to be related to the voltage provided by the source. It is essential for analyzing circuits with several loops, where different paths for the current exist. With this law, equations can be written that show how electrical energy is distributed in each path.

How are Kirchhoff’s Laws applied in a real circuit?

To apply Kirchhoff’s Laws, the first step is to identify the nodes and loops of the circuit. The Current Law is used at the nodes to relate the currents that split among several branches, while the Voltage Law is applied in the loops to connect the voltage drops with the energy supplied by the source. The usual procedure is to choose directions for the currents, write the equations corresponding to each node and each loop, and then solve the resulting system. Although it may seem laborious at first, this method allows complex circuits to be analyzed without reducing them to equivalent resistances, providing all internal currents and voltages based on conservation principles.

What are Kirchhoff’s Laws used for in practice?

Kirchhoff’s Laws are essential for studying circuits that cannot be simplified easily, such as those with several loops, multiple nodes or components arranged in irregular ways. They make it possible to determine unknown currents and voltages anywhere in the circuit, analyze how different branches behave when they share common nodes and check whether a circuit satisfies the conditions of charge and energy conservation. They are also used to verify experimental measurements, design more complex circuits and understand how real electrical networks work, where simplifications are not always possible. Together, they form an indispensable tool for any rigorous electrical analysis.