0.1 Kirchhoff's circuit laws  (Page 2/3)

$$\sum V=0$$

where V stands for potential difference across an element of the circuit.

Applications

Kirchhoff’s laws are extremely helpful in analyzing complex circuits. Their application requires a bit of practice and handful of methods i.e. techniques. Many people like to use a set of procedures which yield results, but they are not intuitive. We shall take a midway approach. We shall rely mostly on the laws as defined and few additional techniques. Some of the useful techniques or procedures are discussed here with examples illustrating the application. The basic idea is to generate as many equations as there are unknowns (current, voltage etc.) to analyze the circuit.

Direction of current (doc)

We assign current direction between two nodes i.e. in the arm in any manner we wish. The solution of the problem will eventually yield either positive or negative current value. A positive value indicates that the assumed direction of current is correct. On the other hand, a negative value simply means that current in that particular arm flows in a direction opposite to assumed direction. See the manner in which current directions are indicated for the same circuit in two different ways :

Direction of currents

Application of KCL to the current assignments in first figure at node C yields :

$$\sum I=I_1+I_2+I_3=0$$

Application of KCL to the current assignments in first figure at node C yields :

$$\sum I=-I_1+I_2-I_3=0$$

Alternatively, we denote currents in different branches such that numbers of unknowns are minimized. We can use KCL to reduce number of variables in the circuit using first figure as :

$$I_3=-\left(I_1+I_2\right)$$

Current in resistors

Further, we should also clearly understand that direction of current (DOC) in a closed loop need not be cyclic. Consider the loop EDCFE in the figure above. Here $I_1$ is clockwise whereas $I_2$ is anticlockwise.

Direction of travel (dot)

We apply Kirchhoff’s voltage law to each of the closed loop. In the figure below, there are two loops ABCFA and EDCFE. We arbitrarily select direction of travel (DOT) either clockwise or counterclockwise. There is no restriction on the choice because a change in the direction changes the sign of voltage drop for all elements, which is equated to zero. Hence, choice of direction of travel does not effect the final equation. We write down voltage drop across various circuit elements moving from a node following DOT till we return to the starting node.

Direction of travel

Voltage across resistor and power source

The sign of voltage drop across a resistor depends on the relative direction of DOC and DOT. Consider the loop EDCFE. Starting from node E (say), we move toward D following clockwise DOT (Direction of Travel). From the direction of current (DOC), it is clear that the end of resistor 5 Ω where current enters is at higher potential than at the end where current exits the resistor. Hence, there is a potential drop, which is indicated by a negative sign. On the other hand, we move in the arm CF from C to F in the opposite direction of the current (COD). Here again, the end of resistor 4 Ω where current enters is at higher potential than at the end where current exits the resistor. Hence, there is a potential gain as we move across resistor from C to F, which is indicated by a positive sign.