Kirchhoff's laws for electrical and magnetic circuits

Kirchhoff's first law

The definition of the first law is as follows: “The algebraic sum of the currents flowing through the node is equal to zero. " You can say in a slightly different form: “How many currents flowed into the node, the same amount flowed out, which indicates the constancy of the current "

A chain node is the point where three or more branches join. The currents in this case are distributed in proportion to the resistances of each branch.

I₁= I₂+ I₃

This form of notation is valid for DC circuits. If we use the first Kirchhoff's law for an alternating current circuit, then instantaneous values ​​are used stresses, denoted by the letter İ and is written in complex form, and the calculation method remains the same:

The complex form takes into account both active and reactive components.

Kirchhoff's second law

If the first describes the distribution of currents in the branches, then the second Kirchhoff's law sounds like this: “The sum of the voltage drops in the circuit is equal to the sum of all EMF. " In simple words, the wording is as follows: “The EMF applied to a section of the circuit will be distributed over the elements of this circuit in proportion to the resistances, i.e. according to Ohm's law ".

Whereas for alternating current it sounds like this: “The sum of the amplitudes of the complex EMF is equal to the sum of the complex voltage drops on the elements ".

Z is the total impedance or complex impedance, it includes both the resistive part and the reactance (inductance and capacitance), which depends on the frequency of the alternating current (in direct current there is only an active resistance). Below are the formulas for the complex resistance of the capacitor and inductance:

Here's a picture to illustrate the above:

Then:

Calculation methods according to the first and second Kirchhoff's laws

Let's get down to practical application of theoretical material. To correctly place the signs in the equations, you need to choose the direction of traversing the contour. Look at the diagram:

We suggest choosing a clockwise direction and marking it in the figure:

The dash-dotted line indicates how to follow the contour when drawing up equations.

The next step is to compose the equations according to Kirchhoff's laws. We use the second one first. We arrange the signs as follows: a minus is put in front of the electromotive force if it is directed counterclockwise arrows (the direction we chose in the previous step), then for the EMF directed clockwise - set minus. We compose for each contour, taking into account the signs.

For the first one, we look at the direction of the EMF, it coincides with the dash-dotted line, we put E1 plus E2:

For the second:

For the third:

The signs for IR (voltage) depend on the direction of the loop currents. Here the rule of signs is the same as in the previous case.

IR is written with a positive sign if the current flows in the direction of the loop bypass. And with a "-" sign, if the current flows against the direction of the loop bypass.

The direction of traversing the contour is a conventional value. It is needed only for the arrangement of signs in the equations, it is chosen arbitrarily and does not affect the correctness of the calculations. In some cases, an unsuccessfully chosen bypass direction can complicate the calculation, but this is not critical.

Consider another chain:

There are as many as four EMF sources, but the calculation procedure is the same, first we choose the direction for drawing up the equations.

Now you need to draw up equations according to the first Kirchhoff's law. For the first node (number 1 on the left in the diagram):

I₃ flows in, and I₁, I₄ follows, hence the signs. For the second:

For the third:

Question: "There are four nodes, but there are only three equations, why? " The fact is that the number of equations of the first Kirchhoff rule is:

N_equations= n_knots-1

Those. equations are only 1 less than nodes, because this is enough to describe the currents in all branches, I advise you to go up to the circuit again and check if all the currents are written in the equations.

Now let's move on to constructing equations according to the second rule. For the first circuit:

For the second circuit:

For the third circuit:

If we substitute the values ​​of real voltages and resistances, then it turns out that the first and second laws are true and fulfilled. These are simple examples; in practice, you have to solve much more voluminous problems.

Output. The main thing when calculating using the first and second Kirchhoff laws is to comply with the rule for drawing up equations, i.e. take into account the directions of current flow and bypassing the circuit for the correct placement of signs for each element chains.

Kirchhoff's laws for a magnetic circuit

In electrical engineering, the calculations of magnetic circuits are also important, both laws have found their application here. The essence remains the same, but the type and values ​​change, let's look at this issue in more detail. First you need to understand the concepts.

The magnetomotive force (MDF) is determined by the product of the number of turns of the coil by the current through it:

F = w * I

Magnetic voltage is the product of the magnetic field strength and the current through the section, measured in Amperes:

U_m= H * I

Or magnetic flux through magnetic resistance:

U_m= Ф * R_m

L is the average length of the section, μ_r and μ₀ - relative and absolute magnetic permeability.

By analogy, we write down the first Kirchhoff's law for a magnetic circuit:

That is, the sum of all magnetic fluxes through the node is zero. Have you noticed that it sounds almost the same as for an electrical circuit?

Then the second Kirchhoff's law sounds like “The sum of the MDS in the magnetic circuit is equal to the sum of U_{M­­ ­­}(magnetic stress).

The magnetic flux is equal to:

For an alternating magnetic field:

It depends only on the voltage across the winding, but not on the parameters of the magnetic circuit.

As an example, consider the following path:

Then the following formula will be obtained for ABCD:

For circuits with an air gap, the following relationships are met:

Magnetic core resistance:

And the air gap resistance (right on the core):

Where S is the area of ​​the core.

In order to fully assimilate the material and visually view some of the nuances of using the rules, we recommend that you familiarize yourself with the lectures that are provided on the video:

The discoveries of Gustav Kirchhoff made a significant contribution to the development of science, especially electrical engineering. With their help, it is quite simple to calculate any electrical or magnetic circuit, currents in it and voltages. We hope you now understand Kirchhoff's rules for electrical and magnetic circuits more clearly.

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• Joule-Lenz law
• The dependence of the resistance of the conductor on temperature
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