Kirchhoff’s Laws help in the construction of complicated circuits containing numerous electrical components seen in everyday life. It also helps in the analysis of any electrical circuits, for example, how much current flows in different areas of an electrical circuit? What was the magnitude of the voltage loss in different regions of the network? What is the current direction in each circuit branch? In this article, we will look at Kirchhoff’s current and voltage laws and how they are used in electrical appliances to calculate the current flowing and voltage drop in various areas of complicated circuits. Read further to find more.

What are Kirchhoff’s Laws?

Kirchoff’s Law: Gustav Robert Kirchhoff was a German physicist born in Russia. His work involved researching electrical conduction. In \(1845,\) he formulated two laws known as Kirchhoff’s Voltage Law (KVL) and Kirchhoff’s Current Law (KCL). They combined known as Kirchhoff’s Circuit law. These laws are used for the analysis of circuits. They help in calculating the flow of current in different streams through the network. 

1. Kirchhoff’s \({{\bf{1}}^{{\bf{st}}}}\) law:- It is also known as Kirchhoff’s Current Law (KCL), and it states that the ”total current or charge entering a junction or node is exactly equal to the total current or charge leaving the node, as no charge is lost at the node”. In other words, the algebraic sum of all the currents entering and leaving a node must be equal to zero.

Note:- Kirchhoff’s current law supports the law of conservation of charge. So we can say that Nord or junction is a point in a circuit that does not act as a source or sink of charge(s).

So,

\(\sum\limits_{k = 1}^n {{I_k}} = 0\)

Where \(n\) is the total number of all the branches at with currents flowing towards or away from the node.

i.e \({I_{{\rm{(exiting) }}}} + {I_{{\rm{(entering) }}}} = 0……..\left( 1 \right)\)

Let us understand this with an example. Focus on node \(A\) from a resistor network. Four branches are connected to this node. There are two incoming currents named \({i_1}\) and \({i_2}\) and two outgoing currents named \({i_3}\) and \({i_4}.\) Now, according to Kirchhoff’s current law, the sum of total incoming and outgoing currents at node \(A\) will be equal to zero.

Now, consider the two currents entering the node, \({i_{1,}}\) and \({i_{2,}}\) with a positive value, and the two currents leaving the node, \({i_3}\) and \({i_4}\) are negative in value. So, we can also rewrite the equation \((1)\) as:

\({i_1} + {i_2} – {i_3} – {i_4} = 0……\left( 2 \right)\)

2. Kirchhoff’s \({{\bf{2}}^{{\bf{nd}}}}\) Law:- It is also known as Kirchhoff’s Voltage Law (KVL), and it states that the“voltage drop around a loop equals to the algebraic sum of the voltage drop across every electrical component connected in the same loop for any closed network and also equals to zero”.

Note:- Kirchhoff’s Voltage Law is based on the law of conservation of energy, because the net change in the energy of a charge, after the charge completes a closed path must be zero.

Let’s take an example to understand Kirchhoff’s Voltage Law. Consider a part of a resistor network with an internal closed loop, as shown in the figure below. We want to write the voltage drops in the closed-loop. According to Kirchhoff’s voltage law, the sum of all the voltage drops across the components connected in the loop \(ABCDA\) is equal to zero.

So we can write:-  

\(\sum\limits_{k = 1}^n {{V_k}} = 0\)

Here, \(n\) is the total number of electrical components in the loop.

i.e \({V_{AB}} + {V_{BC}} + {V_{CD}} + {V_{DA}} = 0\)

How To Apply Kirchhoff’s Laws? 

When applying KCL, we have to consider the currents leaving a junction to be negative and the currents entering the junction to be taken as positive in sign.

Also, during the application of KVL, we maintain the same anti-clockwise or clockwise direction from the point we started in the loop and account for all voltage drops as negative and rises as positive. This leads us to the starting point where the final sum of all the voltage drop is zero.

Sign conventions: 

1. An increase of potential difference or \({\rm{EMF}}\) from lower to higher is always considered positive in a loop. 
2. A decrease of potential difference or \({\rm{EMF}}\) from higher to lower is always considered negative in a loop.
3. The voltage drop across the resistor is taken as negative if the direction of the looping is the same as the direction of the current flowing through the circuit. 

Kirchhoff’s Law is Applicable To?

1. To determine the values of current flowing and voltage drop in the different parts of the complex circuit.
2. It helps in knowing the direction of current in different loops of the circuits.
3. Kirchhoff’s Laws are useful in understanding the transfer of energy through an electric circuit.

Kirchhoff’s Laws Limitations

1. The major drawback of Kirchhoff’s law is that it assumes there is no fluctuating Magnetic field across the area of the loop which can cause a change in magnetic flux and generation of \({\rm{EMF}}\) in the circuit. This may cause an error in the calculation of high-frequency \({\rm{AC}}\) circuits.
2. Kirchhoff also neglected the effect of the electric field produced due to other parts of the circuit.
3. \({\rm{KCL}}\) is applicable on the assumption that current flows only in conductors and wires. While in High-Frequency circuits where parasitic capacitance can no longer be ignored. In such cases, current may startcan flowing in an open circuit because in these cases, conductors or wires are acting as transmission lines.

Kirchhoff’s Laws – Sample Problems

Q.1. From the given circuit in the below image, find the value of \(I\)?

Ans: Apply Kirchhoff’s first law to the point \(P\) in the given circuit.
Let consider the sign convention as the arrows pointing towards \(P\) is positive and away from \(P\) are negative.
Therefore, we have:
\(0.2\,{\rm{A}} – 0.4\,{\rm{A}} + 0.6\,{\rm{A}} – 0.5\,{\rm{A}} + 0.7\,{\rm{A}} – I = 0\)
\( \Rightarrow 1.5\,{\rm{A}} – 0.9\,{\rm{A}} – I = 0\)
\( \Rightarrow 0.6\,{\rm{A}} – I = 0\)
\( \Rightarrow I = 0.6\,\rm{A}\)

Q.2. Use Kirchhoff’s rules to find the value of unknown resistance \(R\) in the below circuit, such that there is no current flowing through \(4\) ohms \(\left( \Omega \right)\) resistance. Also, find the potential difference between points \(A\) and \(D.\)

Ans: Since it is given in the question that there is no current flowing through the \(4\,\Omega \) resistor, so all the current flowing along \(FE\) will go along \(ED\) (By Kirchhoff’s first law).
Then, the current distribution is shown in the below circuit

Now, Applying Kirchhoff’s second law in mesh \(AFEBA,\)
We have:- \( – 1 \times I – 1 \times I – 4 \times 0 – 6 + 9 = 0\)
\(\Rightarrow \,\,\, – 2I + 3 = 0\)
\(\Rightarrow \,\,\,\,I = \frac{3}{2}\,\rm{A}\,\,\,\,\,\,\,\,\,\,\,\,…..\left( {\rm{1}} \right)\)
Again, Applying Kirchhoff’s \({{\rm{2}}^{{\rm{nd}}}}\) law in mesh \(AFDCA,\)
We have: \( – 1 \times I – 1 \times I – I \times R – 3 + 9 = 0\)
\(\Rightarrow \,\,\, – 2I – IR + 6 = 0\)
\( \Rightarrow 2I + IR = 6……..\left( 2 \right)\)
From equations \(\left( 1 \right)\) and \(\left( 2 \right),\) we get
\( \Rightarrow \left( {2 \times \frac{3}{2}} \right) + \frac{3}{2}R = 6\)
\( \Rightarrow R = 2\,\Omega \)
Again, for potential differences across \(A\) and \(D\) along with AFD,
We have:- \({V_A} – \frac{3}{2} \times 1 – \frac{3}{2} \times 1 = {V_D}\)
\( \Rightarrow {V_A} – {V_D} = 3\,\rm{V}\)

Summary

Kirchhoff’s circuit laws are important to circuit analysis. We have the essential instrument to begin studying circuits with the use of these principles and the equations for individual components (resistor, capacitor, and inductor).

1. Gustav Kirchhoff provided a better understanding to solve simple as well as complex circuits and networks.
2. The first law of Kirchhoff states that the total current that enters a node or junction is equal to the total current or charge leaving the node. It is based on Conservation of Charge. This is also called the junction rule.

i.e \(\sum\limits_{k = 1}^n {{I_k}} = 0\)

3. The second law of Kirchhoff states that the sum of voltage drops across each electrical component connected in the loop will be equal to zero. It is based on the law of Conservation of Energy. This is also called the loop rule.                                                                     

i.e \(\sum\limits_{k = 1}^n {{V_k}} = 0\)

4. Kirchhoff’s law is not suitable for high-frequency \({\rm{AC}}\) circuits.

FAQs on Kirchhoff’s Laws

Q.1. What is the Junction and loop Rule?
Ans: The junction rule is also known as Kirchhoff’s Current Law KCL and it states that at any junction the sum of the entering currents is equal to the sum of the leaving currents.
Kirchhoff’s Loop Rule also known as Kirchhoff’s Voltage Law KVL and it states that the sum of the voltage differences around the loop must be equal to zero.

Q.2. What is Node Voltage?
Ans: When we use the term node voltage, we are referring to the potential difference between two nodes of a circuit. We select one of the nodes in the given circuit as a reference node. All the voltages of other nodes are measured concerning this one reference node.

Q.3. What is the importance of Kirchhoff’s law in daily life?
Ans: Kirchhoff’s laws can be used to determine the values of unknown values like current, voltage in the circuit. These laws can be applied in any circuit (with some limitations), and useful to find the unknown values in complex circuits and networks. It helps in knowing the energy transfer in different parts of the circuit.

Q.4. Does Kirchhoff’s law fail at high frequency?
Ans: Yes, Kirchhoff’s laws fail at high frequency, because both the law \({{\rm{KCL}}}\) and \({{\rm{KVL}}}\) are not suitable for \({{\rm{AC}}}\) circuits of high frequencies. At higher frequencies, the interference of induced emf due to varying magnetic fields becomes significant.

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