Easy Kirchhoff’s Circuit Laws for Electric Circuit

For a starting explanation, try to use the calculators below. After that proceed to their complete explanations below them and try to use the calculator again along with your own calculations

Kirchhoff’s Law Calculator

After learning the Kirchhoff’s laws above, we can simplify and shorten our work with the Kirchhoff’s law calculator below that consists of:

1. Kirchhoff’s current law calculator
2. Kirchhoff’s current law calculator

It is not a wise thing to use this calculator before learning how do we use Kichhoff’s circuit laws first.

Kirchhoff’s Voltage Law Calculator

After learning about Kirchhoff’s laws, we can use a simple KVL calculator below to make our work easier.

Kirchhoff’s Current Law Calculator

After learning about Kirchhoff’s laws, we can use a simple KCL calculator below to make our work easier.

If the equations are not displayed correctly, please use the desktop view

Kirchhoff’s Circuit Laws

Kirchhoff’s circuit laws are considered as the fundamental circuit analysis laws and theorems. But don’t get mistaken, these laws are still used even if we use more advanced circuit analysis laws and theorems.

Kirchhoff’s laws were introduced first by a German physicist, Gustav Kirchhoff in 1845. These laws summarized the research of Georg Ohm and James Clerk Maxwell.

Kirchhoff’s laws are introduced to overcome the difficulty of using only Ohm’s law to analyze a circuit. Solving a circuit with multiple voltage or current sources will cost us a lot of time just to collect all the equations.

Leaving aside the strong point of Kirchhoff’s circuit laws, we will still use Ohm’s law to solve the equations we got from Kirchhoff’s laws.

There are few points and terms you need to understand before studying and mastering Kirchhoff’s circuit laws; they are branches, nodes, and loops. You can learn them first before continuing your study here if you haven’t had enough understanding.

The Kirchhoff circuit law is divided into:

• Kirchhoff’s current law
• Kirchhoff’s voltage law

With these two, without a doubt we can solve a complex electrical circuit even with multiple junctions, voltage or current sources, and even bridge networks.

Kirchhoff’s laws are flexible because we can use these in frequency and time domain.

Kirchhoff’s Current Law

Kirchhoff’s current law is often called

• Kirchhoff’s first law
• Kirchhoff’s junction rule
• Kirchhoff’s nodal rule, and
• Kirchhoff’s point rule.

Kirchhoff’s first law is based on the law of conservation of charge that requires that the algebraic sum of charge within a system cannot change. Hence,

Kirchhoff’s current laws (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.

Kirchhoff’s current law or KCL, for short, states that the sum of currents entering a node (or junction) is equal to the sum of currents leaving that node.

The current still has a positive or negative quantity, reflecting their flow direction. We can say if the currents have positive signs, they are entering a node. Otherwise, the currents are leaving a node.

The mathematical equation of KCL is

where :
N = number of branches which connected to the node
i_n = the nth current entering or leaving the node

With this law, currents entering a node can be assumed as positive, while the leaving currents as negative or vice versa.

In order to prove KCL, a set of currents i_k(t), k = 1,2,….., flow into a node, the algebraic sum of current at that node is

Integrating both sides of the equation above makes

where

And

But the law of conservation of electric charge requires an algebraic sum of electric charges at the node must not change.

Thus,

confirming the validity of KCL.

Kirchhoff’s Current Law Formula

Please observe the illustration below to understand how KCL works.

As we have read,

Kirchhoff’s current laws (KCL) states that the algebraic sum of currents entering a node (or a closed boundary) is zero.

Assuming the entering currents have positive sign and leaving currents have negative sign, then

Furthermore, we can rewrite the equation above into

Where

We can conclude the alternative form of KCL as

The KCL equation is the sum of currents entering a node is equal to the currents leaving that node.

For easier explanation, let us imagine some current sources connected together in parallel. The combined current is the algebraic sum of the current supplied by individual sources.

Then, this example can be seen in the circuit below.

The combined or equivalent current source can be found by applying KCL to node a.

and then combined to make a connection as seen below

A circuit cannot contain two different current I₁ and I₂ in series unless I₁ = I₂.

Kirchhoff’s Circuit Laws Examples – KCL

For an example, find the value of i in the circuit below:

We can calculate the current leaving the point a (I) using Ohm’s law:

The current is entering the resistor because the positive sign of the 2Ω resistor is facing the point a. Thus, the circuit becomes

Assume that entering currents are positive, otherwise negative. Then,

Now we need to find I₂,

The value of I₂ is

From this circuit, we have every variable we need to solve the question.

The value of i is

The negative value shows that the current should be in the opposite direction. Then

Kirchhoff’s Voltage Law

Kirchhoff’s voltage law is often called:

• Kirchhoff’s second law,
• Kirchhoff’s second rule,
• Kirchhoff’s mesh rule, and
• Kirchhoff’s loop rule.

Kirchhoff’s second law is based on the principle of conservation of energy, hence

Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero.

The principle of Conservation of Energy means: if the current is moving in a closed-loop, it will reach the point where it started in the first place.

Hence, the initial potential has no voltage drop in the loop. Summary, the voltage drop in a loop is equal to the voltage sources met in the way.

It is important to pay attention to the quantity signs (positive and negative) of the circuit element.

If we write the equation with the wrong signs of the circuit element voltage drop, the calculation can be wrong.

Before moving on, let us learn what voltage drop is first.

Above is the example of voltage drop for a single element. We will use a resistor here for easier explanation.

Let’s say the current I is the same with the positive charge flowing direction, from the left to the right (A to B).

We can say the current is flowing from the positive terminal to the negative terminal.

Because we are using the same direction as the same as the current direction, there will be a drop across the resistor.

The value of the voltage drop will be (-iR).

For the best step, we will pay attention to the polarity direction. The polarity sign of the element will follow the flow direction of the current through it.

Just decide the current flow in clockwise or counterclockwise before starting to write the equation.

Both of them will provide the correct answer, even if the result is in negative signs (it means the current is flowing in the opposite direction).

Kirchhoff’s Circuit Laws Examples – KVL

For better understanding, please take a look at the circuit below.

With the mathematical equation, KVL states

where M is the number of voltages in the loop (or the number of branches in the loop) and v_m is the mth voltage.

The sign on each voltage is the polarity of the terminal encountered first as we travel around the loop.

So, we can start with any branch and go around the loop either in a clockwise direction or counterclockwise.

Assume we start with a clockwise direction then the voltages would be –v₁, +v₂, +v₃, –v₄, and +v₅ in order.

Hence the KVL yields

Rearranging equation gives

Which may be interpreted as

For example for the voltage sources in the circuit below.

The combined or equivalent voltage source in the circuit above is obtained by using the KVL equation.

Using two different voltages in parallel is violating KVL unless the values are the same.

Kirchhoff’s laws will take part on:

• Wye-Delta transformation
• Nodal analysis
• Mesh analysis

Kirchhoff’s Voltage Law Example

For better understanding, observe the circuit below

Since the current arrow indicator facing left, the loop should be counterclockwise. The positive sign of the upper resistor is on the right, and on the left for the bottom resistor. The resistance values in this circuit are ignored.

The KVL equation states that the algebraic sum of all voltages around a closed path (or loop) is zero. Then,

The negative value indicates that the positive sign of the upper resistor should be on the left side and the current is in clockwise direction to make it have a positive value.

Kirchhoff’s Laws Limitations

Kirchhoff’s laws may be considered as the simplest circuit analysis. But, they have their own limitations depending on the type of the circuit.

Below are the limitations of Kirchhoff’s laws:

• KCL is used with the assumption that the current is only flowing in wires and conductors. But, it will be different if we analyze High-Frequency circuits, where the parasitic capacitance can’t be ignored anymore.
• For some cases, currents can flow in an open circuit because conductors and wires are acting as transmission lines.
• KVL is used with the assumption that there is no fluctuating magnetic field linking to the closed-loop. While the presence of changing magnetic fields in a High-Frequency circuit but short-wavelength AC circuit, the electric field is not a conservative vector field.
• Electric field and EMF could be induced and cause the KVL breaks.
• In a transmission line, the electric charge changes over time and violates the KCL.

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