Easy 2 Kirchhoff’s Circuit Laws Theory and Equation

kirchhoff circuit law

Kirchhoff’s laws deal with the calculation of current and voltage in electric circuit analysis.

These laws summarized the research of Georg Ohm and James Clerk Maxwell. Kirchhoff’s laws are flexible because we can use these in frequency and time domain.

Kirchhoff’s Circuit Laws

Kirchhoff’s circuit laws contain the two most important laws for analyzing an electric circuit. These laws analyze the current and voltage in a lumped electrical element of an electrical circuit.

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 flexible because we can use these in frequency and time domain. We need to learn about this fundamental analysis theory.

Kirchhoff’s Laws

Ohm’s law alone is not sufficient to cover the analyzing electric circuit. But, if we pair it with Kirchhoff’s two laws, we get a powerful set of analyzing a large variety of electric circuits.

Kirchhoff’s laws were introduced back when 1824 – 1887 by Gustav Robert Kirchhoff. Thus, these laws are known as Kirchhoff’s current laws (KCL) and Kirchhoff’s voltage laws (KVL).

Kirchhoff’s Current Law

Kirchhoff’s current law is often called as 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

Kirchhoff's current law mathematical equation
(1)

where :
N   = number of branches which connected to the node
in  = 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 ik(t), k = 1,2,….., flow into a node, the algebraic sum of current at that node is

Kirchhoff's current law
(2)

Integrating both sides of Equation.(2) makes

Kirchhoff's current law
(3)

where qk(t) = ∫ ik(t)dt and qT(t) = ∫ iT(t)dt. But the law of conservation of electric charge requires an algebraic sum of electric charges at the node must not change.

Thus, qT(t) = 0 so that iT(t) = 0, confirming the validity of KCL.

Please give attention to Figure.(1) below, an illustrated KCL.

Kirchhoff's current law
Figure 1. Kirchhoff current law

Consider the node in Figure.(1). Applying the KCL makes

Kirchhoff's current law
(4)

since the currents i1i3, and i4 are entering the node, while the i2 and i5 are leaving the node, we can arrange the Equation.(4) to

Kirchhoff's current law
(5)

Equation.(5) is an alternative form of KCL:

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

Take note that KCL also applies to a closed boundary.

Therefore, this can be the generalized case because a node may be assumed as a closed surface shrunk to a point.

Figure.(2) is illustrating a two dimensions boundary where the total currents entering the closed surface is equal to the total current leaving the surface.

Kirchhoff's current law
Figure 2. Kirchhoff current law closed boundary

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 Figure.(3a) and then combined to make a connection in Figure.(3b).

Kirchhoff's current law
Figure 3. Equivalent example of Kirchhoff current law

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

Kirchhoff's current law
(6)

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

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 :

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.

Kirchhoff's voltage law

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 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 the result is in negative signs (it means the current is flowing in the opposite direction).

With the mathematical equation, KVL states

Kirchhoff's voltage law
(7)

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

For better understanding, please take a look at Figure.(4).

Kirchhoff's voltage law
Figure 4. Kirchhoff voltage law

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

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

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

Hence the KVL yields

Kirchhoff's voltage law
(8)

Rearranging equation gives

Kirchhoff's voltage law
(9)

Which may be interpreted as

Kirchhoff's voltage law
(10)

For example for the voltage sources in Figure.(5a),

Kirchhoff's voltage law
Figure 5. Kirchhoff voltage law equivalent circuit

The combined or equivalent voltage source in Figure.(5b) is obtained by using KVL.

Kirchhoff's voltage law
(11)

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

Kirchhoff’s laws will take part on:

  1. Wye-Delta transformation
  2. Nodal analysis
  3. Mesh analysis

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.

Kirchhoff’s Laws Examples

To help you understand better, let us review the examples below.
1.For the circuit in Figure.(6), find voltages v1 and v2.

Kirchhoff's voltage law
Figure 6

Solution :
To find v1 and v2, we apply Ohm’s law and Kirchhoff’s voltage law.
Assume the current is flowing in a clockwise direction.
From Ohm’s law,

v1 = 2iv2= -3i…………..(1.1)

Applying KVL

-20 + v1– v2= 0…………..(1.2)

Substituting equation (1.1) and (1.2) above, we get

-20 + 2i + 3i = 0

or

5i = 20

then

i = 4 A

Substituting i in (1.1) we get

v1 = 8 V and v= – 12 V


2.Determine vo and i in Figure.(7)

Kirchhoff's voltage law
Figure 7

Solution :
We apply KVL around the loop, gives

– 12 + 4i + 2vo – 4 + 6i = 0…………(2.1)

Applying Ohm’s law to 6Ω gives

vo = -6i………..(2.2)

Substituting (2.1) and (2.2) gives

-16 + 10i – 12i = 0 => i = -8 A

and

vo = 48 V

Frequently Asked Questions

What is Kirchhoff’s first law?

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

What is KCL in the circuit?

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.

What is Kirchhoff’s 2nd law?

Kirchhoff’s second law is based on the principle of conservation of energy : Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero.

Why KVL and KCL fail at high frequency?

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.

What is Kirchhoff voltage law formula?

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

What is the loop rule?

The sign on each voltage is the polarity of the terminal encountered first as we travel around the loop. So that, we can start with any branch and go around the loop either a clockwise direction or counterclockwise.

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.