Kirchhoff's Rules

The famous German physicist Gustav Robert Kirchhoff (1824 - 1887), a graduate of the University of Königsberg, being the head of the Department of Mathematical Physics at the University of Berlin, on the basis of experimental data and Ohm's laws received a number of rules that allowed analyzing complex electrical circuits. So Kirchhoff's rules appeared and are used in electrodynamics.

The first (node rule) is, in essence, the law of conservation of charge in combination with the condition that charges are not born and do not vanish in the conductor. This rule applies to nodes of electrical circuits, i.e. Points of a chain in which three or more conductors converge.

If we take as the positive direction of the current in the circuit that goes to the current node, and the one that departs - for negative currents, then the sum of the currents at any node must be zero, because charges can not accumulate at the node:

I = n

Σ Iᵢ = 0,

I = l

In other words, the number of charges that approach the node per unit time will equal the number of charges that leave the given point in the same time period.

The second rule of Kirchhoff is a generalization of Ohm's law and refers to closed contours of a branched chain.

In any closed loop arbitrarily chosen in a complex electrical circuit, the algebraic sum of the products of the current and resistance forces of the corresponding sections of the circuit will be equal to the algebraic sum of the emf in the given circuit:

I = n₁ i = n₁

Σ Iᵢ Rᵢ = Σ Ei,

I = li = l

Kirchhoff's rules are most often used to determine the magnitude of current forces in sections of a complex circuit, when resistances and parameters of current sources are specified. Consider the technique of applying rules on the example of calculating the circuit. Since the equations in which the Kirchhoff rules are used are ordinary algebraic equations, their number must equal the number of unknown quantities. If the analyzed chain contains m nodes and n sections (branches), then according to the first rule it is possible to compile (m - 1) independent equations, and using the second rule, still (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 can not be a source or a sink of charges. If you make a mistake when selecting the direction of the current , then the value of the strength of this current will be negative. But the directions of the action of the current sources are not arbitrary, they are dictated by the way of switching the poles.

Step 2. We write the current equation corresponding to the first Kirchhoff rule for node b:

I₂ - I₁ - I₃ = 0

Step 3. We write down the equations corresponding to the second Kirchhoff rule, but we first choose two independent contours. In this case, there are three possible options: the left contour {badb}, the right contour {bcdb}, and the contour around the entire chain {badcb}.

Since we only need to find three values of the current, we confine ourselves to two circuits. The direction of the bypass does not matter, the currents and EMF are considered positive if they coincide with the direction of the bypass. Let's go around the contour {badb} counter-clockwise, the equation will look like:

I₁R₁ + I₂R₂ = ε₁

The second round we make on the big ring {badcb}:

I₁R₁ - I₃R₃ = ε₁ - ε₂

Step 4. Now we are making a system of equations, which is quite simple to solve.

Using Kirchhoff's rules, it is possible to perform fairly complex algebraic equations. The situation is simplified if the chain contains certain symmetric elements, in this case there may exist nodes with the same potentials and branch circuits with equal currents, which substantially simplifies the equations.

A classic example of such a situation is the problem of determining the forces of currents in a cubic figure composed of identical resistances. Because of the symmetry of the chain, the potentials of the points 2,3,6, as well as the points 4,5,7, will be identical, they can be connected, since this will not change the distribution of currents in terms of the distribution, but the circuit will be much simpler. Thus, the Kirchhoff law for the electrical circuit pivots easily to calculate a complex DC circuit.