Kirchhoff’s circuit law

Kirchhoff’s circuit laws are two laws related to the conservation of charge and energy in electric circuits with concentrated parameters.

In the study of electrical circuits it happens to have to determine some unknown electrical quantities (effects) according to the value assumed by other known electrical quantities (causes). To solve this problem it is necessary to have equations that relate causes to effects. The German scientist Gustav Robert Kirchhoff (1824-1887) derived the two fundamental principles for the study of electrical networks that allow to write such equations. He deduced his principles from two general physical laws that are not unique to electrical networks, as a result of empirical experiments and historically precede the much more complex and general Maxwell equations.

  1. Kirchhoff’s current law (1st Law) states that the current flowing into a node (or a junction) must be equal to the current flowing out of it. This is a consequence of charge conservation.
  2. Kirchhoff’s voltage law (2nd Law) states that in any complete loop within a circuit, the sum of all voltages across components which supply electrical energy (such as cells or generators) must equal the sum of all voltages across the other components in the same loop. This law is a consequence of both charge conservation and the conservation of energy.

First Kirchhoff principle

Consider a node of an electric circuit belonging to a generic electric network. In it converge different branches, each one crossed by an electric current having the conventional direction (indicated in the figure).

It can be reasonably assumed that the total current exiting the node is equal to the total current entering it. On the other hand, if this did not happen we would have a progressive accumulation or rarefaction of electric charges depending on the term that is greater. In practice it is considered constant the total charge present in each part of an electric network, or it is considered valid the principle of stationarity of electric charge.

In stationary regime the electric charge present in the node must remain constant and therefore, in the same time interval, to the charge entering in the node must correspond an equal amount of charge leaving from it. What we have just said is the first Kirchhoff principle.

In a node, the sum of the incoming currents is equal to the sum of the outgoing currents.

Considering positive the incoming currents and negative the outgoing currents the first principle of Kirchhoff can be expressed in the following way:

\[\sum \pm I=0\]

that is, in a node the algebraic sum of currents is zero. Applying Kirchhoff’s first principle to each node of an electrical network we obtain a number of linear equations equal to the number n of nodes. These equations are referred to as the node equations.

Second Kirchhoff principle

The electric field, as well as the gravitational field is conservative. This implies that the variation of potential energy undergone by a test charge q to go from one point to another in the network does not depend on the path followed. Let’s see what this aspect implies. Consider a mesh present inside a generic electric network:

We want to determine the electric potential difference \(V_{AE}\). However, by applying the principle of additivity of potential differences, we can state that:

\[V_{AE} = V_{AB} + V_{BC} + V_{CD} + V_{DE}\]

or, equivalently, choosing the other path:

\[V_{AE} = V_{AG} + V_{GF} + V_{FE}\]

Being the electric field conservative it is evident that the potential difference VAE calculated in the first way is equal to the potential difference calculated in the second way. Putting together the two relationships we get:

\[V_{AB} + V_{BC} + V_{CD} + V_{DE} = V_{AG} + V_{GF} + V_{FE}\]

From which, after simple steps:

\[V_{AB} + V_{BC} + V_{CD} + V_{DE} + V_{EF} + V_{FG} + V_{GA} = 0\]

It is clear that the previously listed voltages cannot all assume only a positive or only a negative value, but, according to the operating regime of the network, some of them will be positive, others will be negative. Therefore the previous summation is not an arithmetic summation but an algebraic one.

We can state in general terms Kirchhoff’s second principle in the following way:

In a Kirchhoff’s loop (or mesh), the algebraic sum of the stresses is zero.

The previous statement can be expressed in a mathematical language by the following analytical expression:

\[\sum \pm V_i =0\]

For each mesh present in an electrical network, Kirchhoff’s second principle allows the determination of an equation, called the mesh equation. To correctly write the mesh equations the following practical rules can be of help:

  • fix a direction of travel of the mesh;
  • consider positive the f.e.m. that have the direction in agreement with the direction of travel and negative the f.e.m. that have the direction discordant with the direction of travel chosen;
  • consider positive the voltage drops given by currents in agreement with the direction of travel and negative the voltage drops given by currents with the direction in agreement with the direction of travel.
  • For example, considering the mesh of the previous figure, using the practical rules just explained and setting the direction of travel clockwise of the mesh, the following equation is obtained:

\[E_1-R_1I_1+E_3-R_3I_3+R_4I_2-E_2+R_2I_2 = 0\]

Solving it, we obtain that:

\[E_1-E_2+E_3 = R_1I_1-R_2I_2+R_3I_3-R_4I_2\]

This expression suggests the following way of stating Kirchhoff’s second principle.

In a mesh, the algebraic sum of the electromotive forces equals the algebraic sum of the potential drops.

The previous statement corresponds to the following analytical expression:

\[\sum \pm E = \pm R I\]

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