The famous German physicist Gustav Robert Kirchhoff (1824 - 1887), a graduate of the University of Koenigsberg, being the head of the Department of Mathematical Physics at the University of Berlin, based on experimental data and Ohm's laws, received a number of rules that allowed us to analyze complex electrical circuits. This is how the Kirchhoff rules appeared and are used in electrodynamics.
The first (knot rule) is, in essence, the law of conservation of charge in combination with the condition that charges do not arise and do not disappear in the conductor. This rule applies to nodes of electrical circuits, i.e. points of the circuit at which three or more conductors converge.
If we take for the positive direction of the current in the circuit, which approaches the current node, and the one that moves away as negative, then the sum of the currents in any node should be zero, because the charges cannot accumulate in the node:
i = n
β Iα΅’ = 0,
i = l
In other words, the number of charges approaching the node per unit time will be equal to the number of charges that leave the given point for the same period of time.
The second Kirchhoff rule is a generalization of Ohm's law and refers to the closed contours of a branched chain.
In any closed circuit, arbitrarily selected in a complex electrical circuit, the algebraic sum of the products of the currents and resistances of the corresponding sections of the circuit will be equal to the algebraic sum of the EMF in this circuit:
i = nβ i = nβ
β Iα΅’ Rα΅’ = β Ei,
i = li = l
Kirchhoff's rules are most often used to determine the magnitude of the current forces in parts of a complex circuit when the resistances and parameters of the current sources are specified. Consider the methodology for applying the rules on the example of circuit calculation. Since the equations in which the Kirchhoff rules are used are ordinary algebraic equations, their number should be equal to the number of unknown quantities. If the analyzed circuit contains m nodes and n sections (branches), then according to the first rule it is possible to compose (m - 1) independent equations, and using the second rule, more (n - m + 1) independent equations.
Step 1. We choose the direction of the currents in an arbitrary way, observing the "rule" of inflow and outflow, the node cannot be a source or sink of charges. If you make a mistake when choosing the direction of the current , then the value of the strength of this current will turn out to be negative. But the directions of action of the current sources are not arbitrary, they are dictated by the method of switching on the poles.
Step 2. We write the current equation corresponding to the first Kirchhoff rule for node b:
Iβ - Iβ - Iβ = 0
Step 3. We write the equations corresponding to the second Kirchhoff rule, but first we choose two independent contours. In this case, there are three possible options: the left path {badb}, the right path {bcdb} and the path around the entire chain {badcb}.
Since it is necessary to find only three values ββof the current strength, we restrict ourselves to two circuits. The direction of the bypass does not matter, currents and EMF are considered positive if they coincide with the direction of the bypass. We go around the {badb} loop counterclockwise, the equation takes the form:
IβRβ + IβRβ = Ξ΅β
The second round will be done along the big ring {badcb}:
IβRβ - IβRβ = Ξ΅β - Ξ΅β
Step 4. Now we compose a system of equations, which is quite simple to solve.
Using the rules of Kirchhoff, one can carry out fairly complex algebraic equations. The situation is simplified if the circuit contains some symmetrical elements, in this case nodes with the same potentials and branches of the circuit with equal currents can exist, which greatly simplifies the equations.
A classic example of such a situation is the problem of determining the current forces in a cubic figure composed of identical resistances. Due to the symmetry of the circuit, the potentials of points 2,3,6, as well as points 4,5,7 will be the same, they can be connected, since this will not change the distribution of currents in terms of, but the circuit will be greatly simplified. Thus, the Kirchhoff law for an electric circuit makes it easy to calculate a complex DC circuit.