Circuits,Electric : Kirchhoff’s Laws

 

Kirchhoff’s current law (KCL)

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

Mathematically, KCL implies that: 

where N is the number of branches connected to the node and  i_n current entering (or leaving) the node. By this la w, currents entering a node may be regarded as positive, while currents leaving the node may be taken as negative or vice versa.
To prove KCL, assume a set of currents i_k (t), k = 1, 2,…, flow into a node. The algebraic sum of currents at the node is: 

Integrating both sides of Eq. (2.14) gives: 

where : 

But the law of conservation of electric charge requires that the algebraic sum of electric charges at the node must not change; that is, the node stores no net charge. Thus, q_T(t) = 0 → i_T(t) = 0, confirming the validity of KCL.
Consider the node in Fig. 2.16. Applying KCL gives:

Figure 2.16 : Currents at a node illustrating KCL

since currents i₁, i₃, and i₄ are entering the node, while currents i₂ and i₅ are leaving it. By rearranging the terms, we get:

Equation (2.17) is an alternative form of KCL:

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

Note that KCL also applies to a closed boundary . This may be regarded as a generalized case, because a node may be regarded as a closed surface shrunk to a point. In two dimensions, a closed boundary is the same as a closed path. As typically illustrated in the circuit of Fig. 2.17, the total current entering the closed surface is equal to the total current leaving the surface. 

Figure 2.17: Applying KCL to a closed boundary.

A simple application of KCL is combining current sources in parallel. The combined current is the algebraic sum of the current supplied by the individual sources. For example, the current sources shown in Fig. 2.18(a) can be combined as in Fig. 2.18(b). The combined or equivalent current source can be found by applying KCL to node a.

Figure 2.18: Current sources in parallel: (a) original circuit, (b) equivalent circuit.

A circuit cannot contain two different currents, I₁ and I₂, in series, unless I₁ = I₂; otherwise KCL will be violated.

Kirchhoff’s voltage law (KVL) 

 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.

where M is the number of voltages in the loop (or the number of branches in the loop) and v_m is the mth voltage. 
To illustrate KVL, consider the circuit in Fig. 2.19.

Figure 2.19: A single-loop circuit illustrating KVL.

The sign on each voltage is the polarity of the terminal encountered first as we travel around the loop. We can start with an y branch and go around the loop either clockwise or counterclockwise. Suppose we start with the voltage source and go clockwise around the loop as shown; then voltages would be −v₁, +v₂, +v₃, +v₄, and −v₅, in that order. For example, as we reach branch 3, the positive terminal is met first; hence, we have +v₃. For branch 4, we reach the negative terminal first; hence, −v₄. Thus, KVL yields

This is an alternative form of KVL. Notice that if we had traveled counter clock wise, the result would have been +v₁, −v₅, +v₄, −v₃, and −v₂, which is the same as before except that the signs are reversed. Hence, Eqs. (2.20) and (2.21) remain the same.

When voltage sources are connected in series, KVL can be applied to obtain the total voltage. The combined voltage is the algebraic sum of the voltages of the individual sources. F or example, for the voltage sources shown in Fig. 2.20(a), the combined or equivalent voltage source in Fig. 2.20(b) is obtained by applying KVL.

Figure 2.20 Voltage sources in series: (a) original circuit, (b) equivalent circuit.

To avoid violating KVL, a circuit cannot contain two different voltages V₁ and V₂ in parallel unless V₁ = V₂.

Example :

For the circuit in Figure, find voltages v₁ and v₂.

Solution:

To find v₁ and v₂ we apply Ohm’s law and Kirchhoff’s voltage law. Assume that current i flows through the loop as shown in Fig. (b). From Ohm’s law, 

Applying KVL around the loop gives: 

Substituting Eq. (2.5.1) into Eq. (2.5.2), we obtain: 

Substituting i in Eq. (2.5.1) finally gives :