Kirchhoff’s Laws

Kirchhoff’s Laws

Posted by Prabhat Jani on 09:13 in Basic Electrical Engg, Circuit Analysis, Electrical Laws

Two laws given by Gustav Robert Kirchhoff (1824–1887) are very useful in writing network equations. These laws are known as Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL). These laws do not depend upon, whether the circuit is made of resistance, inductance or capacitance, or a combination of them.

2.5.1 Kirchhoff’s Current Law

This law is applied at any node of an electric network. This law states that the algebraic sum of currents meeting at a junction or a node in a circuit is zero. KCL can be expressed mathematically as

where n is the number of branches meeting at a node and I_j represents the current in the jth branch as has been shown in Fig. 2.20.

Figure 2.20 (a) Application of Kirchhoff's current law; (b) circuit for application of KVL

By observing Fig. 2.20, we can state KCL in another form:

The sum of current flowing towards a junction or a node is equal to the sum of currents flowing out of the junction.

The current entering the junction has been taken as positive while the currents leaving the junction have been taken as negative. That is to say there is no accumulation of current in a junction.

2.5.2 Kirchhoff’s Voltage Law

This law is applicable to any closed loop in a circuit.

KVL states that at any instant of time the algebraic sum of voltages in a closed loop is zero.

In applying KVL in a loop or a mesh a proper sign must be assigned to the voltage drop in a branch and the source of voltage present in a mesh. For this, a positive sign may be assigned to the rise in voltage and a negative sign may be assigned to the fall or drop in voltage.

KVL can be expressed mathematically as

where V_j represents the voltages of all the branches in a mesh or a loop, i.e., in the jth element around the closed loop having n elements.

Let us apply KCL and KVL in a circuit shown in Fig. 2.20 (b). The current flowing through the branches have been shown.

Applying KCL at node B, we can write

Now, let us apply KVL in mesh ABEFA and mesh CBEDC, respectively.

For the mesh ABEFA, starting from point A, the sum of voltage drops and voltage rise are equated to zero as

or,

The students need to note that while we move in the direction of the flow of current, the voltage across the circuit element is taken as negative. While we move from the negative terminal of the source of EMF to the positive terminal, the voltage is taken as positive. That is why we had taken voltage drop across the branch AB as +I₁R₁ and across BE as −I₃R₃. Since we were moving from the positive terminal of the battery towards its negative terminal while going round the mesh we had considered it as voltage drop and assigned a negative sign.

Using this convention, for the mesh CBEDC, applying KVL we can write

or,

In the two equations, i.e., in (ii) and (iii), if the values of R₁, R₂, R₃, E₁, and E₂, are known, we can calculate the branch currents by solving these equations.

Students need to note that Kirchhoff’s laws are applicable to both dc and ac circuits.

Let us apply KVL in a circuit consisting of a resistance, an inductance, and a capacitance connected across a voltage source as has been shown in Fig. 2.21. We will equate the voltage rise with the voltage drops.

Figure 2.21 Application of KVL

The voltage equation is

While solving network problems using Kirchhoff’s laws we frame a number of simultaneous equations. These equations are solved to determine the currents in various branches in a circuit. We will discuss solving of simultaneous equations by the method of determinants or Cramer’s Rule.