The Kirchhoff 's laws are founded on the law of conservation of energy, and allow us to analyze the variables inherent in electrical circuits. Both precepts were enunciated by the Prussian physicist Gustav Robert Kirchhoff in mid-1845, and are currently used in electrical and electronic engineering to calculate current and voltage.
The first law says that the sum of the currents that enter a node of the circuit must be equal to the sum of all the currents that are expelled from the node. The second law states that the sum of all the positive voltages in a mesh must equal the sum of the negative voltages (the voltage drops in the opposite direction).
Gustav Robert Kirchhoff
Kirchhoff's laws, together with Ohm's Law, are the main tools available to analyze the value of the electrical parameters of a circuit.
Through the analysis of nodes (first law) or meshes (second law) it is possible to find the values of the currents and the voltage drops that occur at any point in the assembly.
The above is valid due to the foundation of the two laws: the law of conservation of energy and the law of conservation of electric charge. Both methods complement each other, and can even be used simultaneously as mutual testing methods for the same electrical circuit.
However, for its correct use it is important to watch over the polarities of the sources and the interconnected elements, as well as the direction of current flow.
A failure in the reference system used can totally modify the performance of the calculations and provide a wrong resolution to the circuit analyzed.
Kirchhoff's First Law
Kirchhoff's first law is based on the law of conservation of energy; more specifically, in balancing the flow of current through a node in the circuit.
This law is applied in the same way in circuits of direct and alternating current, all based on the law of conservation of energy, since energy is neither created nor destroyed, it is only transformed.
This law establishes that the sum of all the currents that enter a node is equal in magnitude with the sum of the currents that are expelled from said node.
Therefore, electric current cannot appear out of nowhere, everything is based on the conservation of energy. The current entering a node must be distributed among the branches of that node. Kirchhoff's first law can be expressed mathematically as follows:
That is, the sum of the incoming currents to a node is equal to the sum of the outgoing currents.
The node cannot produce electrons or deliberately remove them from the electrical circuit; that is, the total flow of electrons remains constant and is distributed through the node.
Now, the distribution of currents from a node can vary depending on the resistance to the flow of current that each derivation has.
Resistance is measured in ohms, and the greater the resistance to current flow, the lower the intensity of the electric current flowing through that shunt.
Depending on the characteristics of the circuit, and on each of the electrical components that make it up, the current will take different paths of circulation.
The flow of electrons will find more or less resistance in each path, and this will directly influence the number of electrons that will circulate through each branch.
Thus, the magnitude of the electric current in each branch can vary, depending on the electrical resistance that is present in each branch.
Example
Next we have a simple electrical assembly in which we have the following configuration:
The elements that make up the circuit are:
- V: 10 V voltage source (direct current).
- R1: 10 Ohm resistance.
- R2: 20 Ohm resistance.
Both resistors are in parallel, and the current inserted into the system by the voltage source is bifurcated towards the resistors R1 and R2 at the node called N1.
Applying Kirchhoff's Law we have that the sum of all the incoming currents in node N1 must be equal to the sum of the outgoing currents; thus, we have the following:
It is known beforehand that, given the configuration of the circuit, the voltage in both branches will be the same; that is, the voltage provided by the source, since it is two meshes in parallel.
Consequently, we can calculate the value of I1 and I2 by applying Ohm's Law, whose mathematical expression is the following:
Then, to calculate I1, the value of the voltage provided by the source must be divided by the value of the resistance of this branch. Thus, we have the following:
Analogous to the previous calculation, to obtain the circulating current through the second derivation, the source voltage is divided by the value of the resistance R2. In this way you have to:
Then, the total current supplied by the source (IT) is the sum of the magnitudes previously found:
In parallel circuits, the resistance of the equivalent circuit is given by the following mathematical expression:
Thus, the equivalent resistance of the circuit is as follows:
Finally, the total current can be determined through the quotient between the source voltage and the total equivalent resistance of the circuit. So:
The result obtained by both methods coincides, with which a practical use of Kirchhoff's first law is demonstrated.
Kirchhoff's Second Law
Kirchhoff's second law indicates that the algebraic sum of all voltages in a closed loop or mesh must equal zero. Expressed mathematically, Kirchhoff's second law is summarized as follows:
The fact that it refers to the algebraic sum implies taking care of the polarities of the energy sources, as well as the signs of the voltage drops on each electrical component of the circuit.
Therefore, when applying this law, one must be very careful in the direction of current flow and, consequently, with the signs of the voltages contained within the mesh.
This law is also based on the law of conservation of energy, since it is established that each mesh is a closed conductive path, in which no potential is generated or lost.
Consequently, the sum of all the voltages around this path must be zero, to honor the energy balance of the circuit within the loop.
Law of conservation of charge
Kirchhoff's second law also obeys the law of conservation of charge, since as electrons flow through a circuit, they pass through one or more components.
These components (resistors, inductors, capacitors, etc.), gain or lose energy depending on the type of element. This is due to the elaboration of a work due to the action of microscopic electric forces.
The occurrence of a potential drop is due to the execution of work within each component in response to the energy supplied by a source, either direct or alternating current.
In an empirical way -that is, thanks to results obtained experimentally-, the principle of conservation of electric charge establishes that this type of charge is neither created nor destroyed.
When a system is subjected to interacting with electromagnetic fields, the related charge on a mesh or closed loop is fully maintained.
Thus, when adding all the voltages in a closed loop, considering the voltage of the generating source (if it is the case) and the voltage drops over each component, the result must be zero.
Example
Analogous to the previous example, we have the same circuit configuration:
The elements that make up the circuit are:
- V: 10 V voltage source (direct current).
- R1: 10 Ohm resistance.
- R2: 20 Ohm resistance.
This time the closed loops or meshes of the circuit are emphasized in the diagram. These are two complementary ties.
The first loop (mesh 1) is made up of the 10 V battery located on the left side of the assembly, which is in parallel with resistor R1. For its part, the second loop (mesh 2) is made up of the configuration of the two resistors (R1 and R2) in parallel.
Compared to the example of Kirchhoff's first law, for the purposes of this analysis it is assumed that there is a current for each mesh.
At the same time, the direction of current flow is assumed as a reference, determined by the polarity of the voltage source. That is, it is considered that the current flows from the negative pole of the source towards the positive pole of this.
However, for the components the analysis is opposite. This implies that we will assume that the current enters through the positive pole of the resistors and leaves through the negative pole of the resistor.
If each mesh is analyzed separately, a circulating current and an equation will be obtained for each of the closed loops in the circuit.
Starting from the premise that each equation is derived from a mesh in which the sum of the voltages is equal to zero, then it is feasible to equal both equations to solve for the unknowns. For the first mesh, the analysis by Kirchhoff's second law assumes the following:
The subtraction between Ia and Ib represents the actual current flowing through the branch. The sign is negative given the direction of flow of the current. Then, in the case of the second mesh, the following expression is derived:
The subtraction between Ib and Ia represents the current that flows through said branch, considering the change in the direction of circulation. It is worth highlighting the importance of algebraic signs in this type of operation.
Thus, by equating both expressions - since the two equations are equal to zero - we have the following:
Once one of the unknowns has been cleared, it is feasible to take any of the mesh equations and solve for the remaining variable. Thus, when substituting the value of Ib in the equation of mesh 1 we have:
When evaluating the result obtained in the analysis of Kirchhoff's second law, it can be seen that the conclusion is the same.
Starting from the principle that the current that circulates through the first branch (I1) is equal to the subtraction of Ia minus Ib, we have:
As you can see, the result obtained by implementing the two Kirchhoff laws is exactly the same. Both principles are not exclusive; on the contrary, they are complementary to each other.
References
- Kirchhoff's Current Law (nd). Recovered from: electronics-tutorials.ws
- Kirchhoff's Laws: Physics Concept (nd). Recovered from: isaacphysics.org
- Kirchhoff's Voltage Law (nd). Recovered from: electronics-tutorials.ws.
- Kirchhoff's Laws (2017). Recovered from: electrontools.com
- Mc Allister, W. (nd). Kirchhoff's Laws. Recovered from: khanacademy.org
- Rouse, M. (2005) Kirchhoff's Laws for current and voltage. Recovered from: whatis.techtarget.com