Kirchhoff Laws

In these revision notes for Kirchhoff Laws, we cover the following key points:

• What are junctions, nodes, paths, branches and loops in a circuit?
• What happens to currents in a node of the circuit?
• What happens to the electromotive force when two sources are installed in the opposite way in a conducting wire?
• How the energy conservation law is applied in complex circuits?
• What procedure do we use to solve complex circuits?

Kirchhoff Laws Revision Notes

1. A junction is a point in which two or more conducting wires converge.
2. A node is a point where two or more circuit elements' terminals are connected together.
   There are two types of nodes: simple node that connects two elements of the circuit that are in the same conducting wire while when the node is located in the junction of two or more wires, it is called a principal node.
3. A path is the set of wires, components and nodes through which the current flows, where all the above elements are counted only once.
4. A branch is the path between one node and another node.
5. A loop is any closed path in a circuit.

In other words, a loop is a closed path starting from a node passing through a set of nodes and returning to the starting node without passing the same node more than once.

If the application of Ohm's Law is impossible due to the complexity of circuits, we use two laws known as Kirchhoff Laws to find the missing quantities in the circuit. They are:

Kirchhoff First Law (the Law of Currents) states that:

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

The mathematical equation that expresses the First Kirchhoff Law (or the Kirchhoff Law of Currents) is

The Kirchhoff's Second Law (the Law of Voltages) states that

The algebraic sum of all the voltages around any closed loop in a circuit is equal to zero.

This law is true because a circuit loop is a closed conducting path and therefore, no energy is lost. The charge flowing through a closed loop is also constant. Recall the relationship between electric potential energy and potential difference

The mathematical form of the Kirchhoff's Voltage Law is

Ohm's Law is just a special case of Kirchhoff's Law of Voltages because in a single resistor and single source circuit (if not considering the resistances of wire and source) we have

The Kirchhoff Law of Voltages is particularly useful when there is more than source in a single branch, especially when they are connected in opposite directions.

The procedure for solving a circuit using the Kirchhoff Laws is as follows:

Step 1 - Make sure to write a clear circuit diagram on which you can label all known and unknown resistances, electromotive forces, and currents. If you are not sure about the direction of any current, you must anyway assign it a direction. This is necessary for determining the signs of potential differences. If you assign the direction of current incorrectly, it will result in a negative value. This is not a problem; you just realize that current is flowing in the opposite direction.

Step 2 - Apply the current rule in each principal node (junction). Every time you must get different equations, otherwise the equations are redundant, i.e. they repeat themselves. If there are only two opposite principal nodes in the circuit, the equation of currents is written only once, as the other is redundant.

Step 3 - Apply the loop rule for as many loops as needed (not necessary for each loop) to solve for the unknowns in the problem. (There must be as many independent equations as unknowns.) To apply the loop rule, you must choose a direction to go around the loop. Then carefully and consistently determine the signs of the potential differences and electromotive forces for each element.

Step 4 - Solve the system of linear equations. You need to express one of the currents in terms of the other two and substitute it in the system of equations involving voltages and electromotive forces.

Step 5 - Check the results you obtained for the currents substituting them in the equation of the current law. Also, make sure any resistance is not negative or it does not have an unreasonable value (very large or very small).