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Kirchhoff’s circuit 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 in 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 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

(1)

where :

N = number of branches which connected to the node

i_{n} = the *n*th 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

(2)

Integrating both sides of Equation.(2) makes

(3)

where *q _{k}*(t) = ∫

*i*(t)dt and

_{k}*q*(t) = ∫

_{T}*i*(t)dt. But the law of conservation of electric charge requires an algebraic sum of electric charges at the node must not change.

_{T}Thus, *q _{T}*(t) = 0 so that

*i*(t) = 0, confirming the validity of KCL.

_{T}Please give attention to Figure.(1) below, an illustrated KCL.

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

(4)

since the currents i1, i3, and i4 are entering the node, while the i2 and i5 are leaving the node, we can arrange the Equation.(4) to

(5)

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

The

sum of currents enteringa node is equal to thesum of the currentsleaving 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) illustrates a two dimensional boundary where the total currents entering the closed surface is equal to the total current leaving the surface.

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).

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

(6)

A circuit cannot contain two different current *I*_{1} and *I*_{2} in series unless *I*_{1} = *I*_{2}.

**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.

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).

With the mathematical equation, KVL states

(7)

where ** M** is the number of voltages in the loop (or the number of branches in the loop) and

*v*is the

_{m}*m*th voltage.

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

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 –v_{1}, +v_{2}, +v_{3}, –v_{4}, and +v_{5} in order.

Hence the KVL yields

(8)

Rearranging equation gives

(9)

Which may be interpreted as

(10)

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

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

(11)

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

Kirchhoff’s laws will take part on:

**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 *v*_{1} and *v*_{2}.

Solution :

To find *v*_{1} and *v*_{2}, we apply Ohm’s law and Kirchhoff’s voltage law.

Assume the current is flowing in a clockwise direction.

From Ohm’s law,

(1.1)

Applying KVL

(1.2)

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

Substituting *i* in (1.1) we get

2.Determine *v _{o}* and

*i*in Figure.(7)

Solution :

We apply KVL around the loop, gives

(2.1)

Applying Ohm’s law to 6Ω gives

(2.2)

Substituting (2.1) and (2.2) gives

and

**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 do 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 the 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, we can start with any branch and go around the loop either in 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.