Definition of Kirchhoff’s law

In our daily lives, we use electrical and electronic devices immensely. What powers or drives these devices are the electrical circuits. Electrical circuits are complicated. We have to send the exact current in various branches or the precise value of potential difference across the different components for the proper functioning of the devices. For solving these circuits, Kirchhoff proposed two laws. With the help of these two laws, one can not only design a circuit, but also find the current and the voltage distribution across a given circuit.

Gustav Kirchhoff, a German physicist, described these laws in 1845. These laws are accurate for DC circuits and the low-frequency alternating current (AC) circuits. The two laws are known as The Junction Law (or the Current Law) and The Loop Law (or the Voltage Law). With the help of these laws, we can find the current and voltage in an electrical circuit. Let’s take a look at both of these laws- 

• The Junction Law – It states that — “The algebraic sum of all currents in a network of conductors meeting at a point/junction is zero.” 

∑I = 0

Sign Convention – The current entering the junction increases the amount of charge at the point. Thus, it is taken as positive. Otherwise, if the current leaves the junction, it will lead to a loss. So then it will be taken as negative. As per the law, the sum of these positive and negative values should be equal to 0. This law is based on the concept of “Conservation of Charge”. Look at the diagram below to get a better idea.

Here ∑I  = I1+I2-I3+I4-I5-I6

Now, according to the junction law, 

  I1+I2-I3+I4-I5-I6 = 0

or  I1+I2+I4 = I3+I5+I6

or total current entering = total current leaving

There is no active power source in a junction like a battery or a cell. Thus, the junction can neither accumulate nor supply any electric charge. This implies that the total charge entering the junction should be equal to the charge leaving the junction. This is known as Kirchhoff’s first law. It is used along with Ohm’s law to perform nodal analysis.

• The Loop Law – This law is also known as the Voltage Law. It states that “The directed sum of the potential differences around any closed loop is zero”. 

∑Vk = 0

• It also states that the algebraic sum of the product of resistance and current in each part of the loop is equal to the algebraic sum of the emfs in that loop. This law is based on the conservation of voltage. 

∑IR = ∑ε

Sign convention – Talking about sign conventions, if we are moving along the current’s direction, then the potential difference is considered to be positive. Otherwise, it will be taken as negative. The examples below clear the concept of the loop law.

In fig (a), the current flows in an anti-clockwise direction. Thus, the total emf will be 1+2. 

Considering I is the total current, the equation will be,

1+2 = IR1+IR2+IR3+IR4

In fig(b), the cells give out current in opposite directions. Therefore, the net emf will be 1-2(assuming 1>2).  So the equation here will be,

1-2 = IR1+IR2+IR3+IR4

Let us look at an example regarding Kirchhoff’s laws.

Q) In the above-given circuit, find the current passing through the two cells and describe the direction of the currents.

Ans)

First, name the circuit as ABCDEF to help with the naming of the respective loop. Consider I1 to be the current coming out from 6V cell and I2 from the 2V cell.  First Law. The circuit equations then made are as follows – 

For loop ABCDA,

2I1- I2 = 6 – 2

or 2I1-I2 = 4

For loop CDEFC,

I2+6(I1+I2) = 2

or 6I1+7I2 = 2

On solving both of the above equations, we get, 

I1 = 1.5A and I2 = -1A

Since I1 is positive, it means that the 6V cell is supplying 1.5A current to the circuit (leaving the 6V cell). However, I2 is negative. This means that the current is entering the 2V cell. 

Limitations of Kirchhoff’s Law – 

1.  Both of these laws cannot be used for high-frequency AC circuits. 

2.  KVL cannot be applied if there is a changing magnetic field within the closed circuit.

Conclusion –  

Kirchhoff’s Laws are an indispensable part of the fundamentals of circuit analysis in the study of electrical engineering. KCL helps determine the unknown values of charge by using the simple concept of ‘Conservation of charge’. KVL helps determine the resistances and impedance of different circuits, ranging from simple to complex.