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# Kirchhoff’s Laws and its applications

Kirchhoff’s circuit laws are two equations first published by Gustav Kirchhoff in the year 1845. These laws usually address the conservation of energy and charge in the context of electrical circuits. Kirchhoff’s Laws can be derived from the equations of Maxwell. Kirchhoff’s laws are very important for the analysis of closed circuits.

## Kirchhoff’s First Law – The Current Law, (KCL)

Kirchhoffs Current Law or also called KCL states that the “total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node”. This idea by Kirchhoff is also known as the Conservation of Charge.

## Kirchhoffs Current Law

Here, the three currents entering the node, I1, I2, I3 are all positive in value, and the two currents leaving the node, I4 and I5 are negative in value.

I1 + I2 + I3 – I4 – I5 = 0

## Kirchhoff’s Second Law – The Voltage Law, (KVL)

Kirchhoffs Voltage Law or also called KVL states that “in any closed-loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop” which is also equal to zero. The algebraic sum of all voltages in a closed circuit within the loop must be equal to zero. This idea is also known as the Conservation of Energy.

If we are starting at any point in the loop continue in the same direction by noting the direction of all the voltage drops, can be either positive or negative and returning back to the same starting point again. It is very important to maintain the same direction or the final voltage sum will not be equal to zero.

When we are analyzing DC or AC circuits by using Kirchhoffs Circuit Laws, a number of definitions and terminologies are used to describe the parts of the circuit being analyzed such as nodes, paths, branches, loops, and meshes.

Conservation of energy—the principle that energy is neither created nor destroyed—is a ubiquitous principle across many studies in physics, including circuits. Applied to circuitry, it is implicit that the directed sum of the electrical potential differences (voltages) around any closed network is equal to zero. In other words, the sum of the electromotive force (emf) values in any closed loop are equal to the sum of the potential drops in that loop (which may come from resistors).

Another statement for this law is that the algebraic sum of the products of resistances of conductors in a closed loop is equal to the total electromotive force available in that loop.

### Common DC Circuit Theory Terms:

Circuit – a circuit is a closed path or a loop conducting path in which an electrical current flows.

Node – a node is a function within a circuit where two or more circuit elements are connected together giving a connection point between two or more branches. A node is usually indicated by a dot.

Branch – a branch is a single or group of components in a circuit such as resistors or a source which are connected between two nodes.

Mesh – a mesh is a single closed loop series path that does not contain any other paths. There are no loops inside a mesh.

## Limitation

Kirchhoff’s junction law is limited in its applicability. Practically, this is always true so long as the law is applied for a specific point. In a region, the charge density may not be constant. A charge is conserved. The only way in which this is seen is if there is a flow of charge across the boundary. This flow would be current, and thus it violates the Kirchhoff’s junction law. Kirchhoff’s rules are also used to analyze any closed circuit by modifying them for those circuits with electromotive forces, resistors, capacitors, and more. Practically speaking, however, the rules are only useful for characterizing those circuits that cannot be simplified by combining elements in series and parallel.

Kirchhoff’s rules are more applicable and should be used to solve problems involving complex circuits that cannot be simplified by combining circuit elements in series or parallel.

When the released photons reach another surface, they may either be absorbed, reflected, or transmitted. The behavior of a surface with incident radiation is described by the following quantities:

absorptivity (aa) is the fraction of incident radiation absorbed

reflectivity (RR) is the fraction of incident radiation reflected

transmissivity (tt) is the fraction of incident radiation transmitted.

The energy conservation gives a+r+t=1a+r+t=1. For opaque objects, transmissivity t=0t=0 and hence a+r=1a+r=1. A blackbody has absorptivity a=1a=1 and absorbs all radiation incident on it. Its reflectivity is r=0r=0.

The measurement of this is the emissivity (ee) defined bye=E/Eb,e=E/Eb, where EE is radiated power per unit area from the real body at temperature TT, and EbEb is radiated power per unit area from a black-body at the same temperature TT. The emissivity of the blackbody is 1 and that of a real body is between 0 and 1.

Consider a small real body in thermal equilibrium with its surrounding blackbody cavity i.e., Tbody=TcavityTbody=Tcavity. The power emitted per unit area of the blackbody (cavity) is Eb Eb. Thus, the incident power per unit area of the real body is Eb Eb. The power absorbed per unit area of the real body is aEbaEb. The power emitted per unit area of the real body is equal to E=eEbE=eEb. Since real body is in thermal equilibrium with its surrounding, energy balance gives eEb=aEbeEb=aEb i.e, a=ea=e. The relation a=ea=e is known as Kirchhoff’s Law of radiation. It implies that good radiators are good absorbers. Note that Kirchhoff’s law is valid only when the body is in thermal equilibrium with its surrounding.