Metal parts in a car or various electrical devices have the ability to move in a magnetic field and intersect with power lines. Thanks to this, self-induction is formed. We propose to consider anomalous Foucault eddy currents, air flows, their definition, application, influence and how to reduce eddy current losses in a transformer.

Faraday's law states that a change in magnetic flux produces an induced electric field even in empty space.

If a metal plate is inserted into this space, the induced electric field causes an electric current to flow through the metal. These induced currents are called eddy currents.

Foucault currents are flows, the induction of which is carried out in the conducting parts of various electrical appliances and machines, Foucault stray currents are especially dangerous for the passage of water or gases, because. their direction cannot be controlled in principle.

If induced counter currents are created by changing magnetic field, then the eddy currents will be perpendicular to the magnetic field, and their movement will be in a circle if this field is uniform. These induced electric fields are very different from the electrostatic electric fields of point charges.

Practical application of eddy currents

Eddy currents are useful in industry to dissipate unwanted energy, such as the swing arm of a mechanical balance, especially if the current is very high. The magnet at the end of the support sets up eddy currents in a metal plate attached to the end of the bracket, say ansys.

Vortex flows, as physics teaches, can also be used as an effective braking force in transit train engines. Electromagnetic devices and mechanisms on the train near the rails are specially configured to create eddy currents. Thanks to the movement of current, a smooth descent of the system is obtained and the train stops.

Swirling currents are harmful in instrument transformers and for humans. A metal core is used in a transformer to increase the flux. Unfortunately, eddy currents generated in the armature or core can increase energy loss. By constructing a metal core of alternating layers of energy-conducting and non-conducting materials, the size of the induced loops is reduced, thus reducing energy loss. The noise that the transformer produces during operation is a consequence of precisely this design solution.

Video: Foucault eddy currents

Another interesting use vortex waves - their use in electricity meters or medicine. At the bottom of each counter is a thin aluminum disc that is always rotating. This disk moves in a magnetic field, so there are always eddy currents there, the purpose of which is to slow down the movement of the disk. Thanks to this, the sensor works accurately and without fluctuations.

Swirls and skin effect

In the case when very strong eddy currents arise (at high-frequency current), the current density in bodies becomes significantly less than on their surfaces. This is the so-called skin effect, its methods are used to create special coatings for wires and pipes, which are developed specifically for eddy currents and tested under extreme conditions.

This was proven by the scientist Eckert, who studied EMF and transformer installations.

Eddy current principles

A copper wire coil is a common method for reproducing eddy current induction. Alternating current passing through the coil creates a magnetic field in and around the coil. The magnetic fields form lines around the wire and connect to form larger loops. If the current increases in one loop, the magnetic field will expand through some or all of the wire loops that are in close proximity. This induces a voltage in adjacent hysteresis loops, and causes a flow of electrons, or eddy currents, in the electrically conductive material. Any defect in the material, including changes in wall thickness, cracks, and other discontinuities, can alter the flow of eddy currents.

Ohm's law

Ohm's law is one of the most basic formulas to determine electric flow. Voltage divided by resistance, ohms, determines electric current, in amperes. It must be remembered that there is no formula for calculating currents; it is necessary to use examples of calculating the magnetic field.

Inductance

Alternating current passing through the coil creates a magnetic field in and around the coil. As the current increases, the coil induces circulation (eddy) currents in the conductive material located next to the coil. The amplitude and phase of the eddy currents will vary depending on the coil load and its resistance. If a discontinuity occurs in the electrically conductive material at or below the surface, the flow of eddy currents will be interrupted. To set it up and control it, there are special devices with different channel frequencies.

Magnetic fields

The photo shows how eddy electric currents form a magnetic field in a coil. The coils, in turn, create eddy currents in the electrically conductive material and also create their own magnetic fields.

Flaw detection

Changing the voltage across the coil will affect the material, scanning and studying eddy currents allows the production of a device for measuring surface and subsurface discontinuities. Several factors will influence what deficiencies may be found:

1. The conductivity of the material has a significant effect on the path of eddy currents;
2. The permeability of a conductive material also has a huge impact due to its ability to be magnetized. Flat surface much easier to scan than an uneven one.
3. Penetration depth is very important in controlling eddy currents. A surface crack is much easier to detect than a sub-surface defect.
4. The same applies to surface area. How smaller area– the faster the formation of eddy currents occurs.

Detecting a contour with a flaw detector

There are hundreds of standard and custom probes that are manufactured for specific types of surfaces and contours. Edges, grooves, contours, and thickness of the metal contribute to the success or failure of a test. A coil that is located too close to the surface of the conductive material will have best chance to detect ruptures. For complex circuits, the coil is inserted into a special block and attached to the fittings, which allows current to pass through it and monitor its condition. Many devices require special probe and coil moldings to accommodate irregular shape details. The coil may also have a special (universal) shape to fit the design of the part.

Reducing eddy currents

In order to reduce the eddy currents of the inductors, it is necessary to increase the resistance in these mechanisms. In particular, it is recommended to use liquefied wire and insulated wires.

Tokami Fuko(or eddy currents) are currents of inductive nature that appear in massive conductors in an alternating magnetic field. Closed circuits of eddy currents appear deep in the conductor itself. The electrical resistance of a massive conductor is small, therefore, Foucault currents can reach of great importance. The strength of eddy currents depends on the shape and properties of the conductor material, the direction of the alternating magnetic field, the speed at which it changes magnetic flux. The distribution of Foucault currents in a conductor can be very complex.

The amount of heat released per $1 s$ by Foucault currents is proportional to the square of the frequency of change of the magnetic field.

According to Lenz's law, Foucault's currents choose such directions in order to influence the cause that causes them. This means that if a conductor moves in a magnetic field, then it must experience strong braking, which is caused by the interaction of Foucault currents and the magnetic field.

Let us give an example of the emergence of Foucault's shackles. Let us make a copper disk with a diameter of $5 cm$ and a thickness of $6 mm$ fall in a narrow gap between the poles of an electromagnet. If the magnetic field is turned off, the disk falls quickly. Let's turn on the electromagnet. The field should be large (about $0.5T$). The fall of the disk will become slow and will resemble movement in a very viscous medium.

Application of Foucault currents

Toki Fuko play useful role in the rotor asynchronous motor, which is given in rotational movement magnetic field. The very implementation of the operating principle of an asynchronous motor requires the appearance of Foucault currents.

Foucault currents are used to dampen moving parts of galvanometers, seismographs and a number of other instruments. So, a plate - a conductor in the form of a sector - is installed on the moving part of the device. It is introduced into the gap between the poles of the strong permanent magnet. When the plate moves, Foucault currents appear in it, which causes inhibition of the system. Moreover, braking appears only when the plate moves. Therefore, this kind of calming device does not interfere with the system's precise arrival in a state of equilibrium.

The heat released by Foucault currents is used in heating processes. Thus, melting metals using Foucault currents is very advantageous in comparison with other heating methods. The so-called induction furnace is a coil through which current flows. high frequency And great strength. A conductive body is placed inside the coil, and high-intensity eddy currents appear in it, which heat the substance until it melts. This is how metals are melted in a vacuum, which leads to the production of high-purity materials.

When using Foucault currents, the internal metal parts of vacuum installations are heated in order to degas them.

Problems that eddy currents cause. Skin effect

Foucauldian currents can play more than just a useful role. Eddy currents are conduction currents, and part of the energy is dissipated to release Joule heat. Such energy, for example, in the rotor of an asynchronous motor, which is usually made of ferromagnets, heats the cores, thereby deteriorating their characteristics. To combat this phenomenon, cores are produced in the form of thin plates that are separated thin layers insulator and install the plates so that the Foucault currents are directed across the plates. With a small thickness of the plates, eddy currents have a small bulk density. With the advent of ferrites and substances with high magnetoresistance, it became possible to manufacture solid cores.

Eddy currents occur in wires that carry alternating currents, and the direction of the Foucault currents is such that they weaken the current inside the wire and strengthen it near the surface. Consequently, the rapidly changing current is distributed unevenly across the cross-section of the wire. This phenomenon is called skin - effect(surface effect). Because of this phenomenon inner part conductor becomes useless in circuits with high frequency use tubes as conductors. The skin effect can be used to heat the surface layer of a metal, which makes it possible to use this phenomenon for hardening the metal, and by changing the frequency of the field, hardening can be carried out at any required depth.

Approximate formulas that can describe the skin effect in a homogeneous cylindrical conductor:

Figure 1.

where $R_w$ is the effective resistance of a conductor with radius $r$ to alternating current with cyclic frequency $w$. $R_0$ - resistance of the conductor to direct current.

where the effective penetration depth of alternating current ($\delta $) (the distance from the surface of the conductor at which the current density decreases by $e=2.7\$ times compared to the density on its surface) is equal to:

$\mu $ - relative magnetic permeability, $(\mu )_0$ - magnetic constant, $\sigma $ - specific conductivity of the conductor for DC. The thicker the conductor, the more significant the skin effect, the smaller the values ​​of $w$ and $\sigma$ at which it should be taken into account.

Example 1

Exercise: In an experiment with a centrifugal machine, a massive copper disk was attached to it, this disk was brought into rotation with high speed. A magnetic needle was suspended above the disk (without contact). What will happen to the arrow, why?

Solution:

The magnetic needle acts as a magnet that creates a magnetic field, in which the copper conductor rotates. Consequently, in the conductor there arise induced currents- Foucault currents. According to Lenz's rule, eddy currents, interacting with the magnetic field, tend to stop the rotation of the disk or, in accordance with Newton's third law, drag the magnetic needle along with them. This means that the magnetic needle, which hangs above the disk, will turn after it and spin the suspension (thread).

Answer: The magnetic needle will rotate, the reason is eddy currents.

Example 2

Exercise: Explain why the underground cable through which alternating current is transmitted cannot be laid close to metal gas and water pipes?

Solution:

Under the influence of alternating current, an alternating magnetic field appears around the cable; if a conductor (metal pipe) gets into this field, then inductive eddy currents will arise. These currents cause corrosion of metal pipes. In addition, the presence of currents in pipes is dangerous, as there is a possibility of electric shock.

Example 3

Exercise: The pendulum, made of thick sheet copper, has the shape of a truncated sector. It is suspended on a rod and can perform free vibrations around horizontal axis in a magnetic field between the poles of a strong electromagnet. In the absence of a magnetic field, the pendulum oscillates with virtually no damping. Describe the oscillations of a pendulum in the magnetic field of an electromagnet. How can a pendulum be made to oscillate almost without damping in the presence of a magnetic field?

Solution:

If the described massive pendulum oscillating is placed in a strong magnetic field, then Foucault currents arise in the pendulum. These currents, according to Lenz's rule, slow down the movement of the pendulum, the amplitude of the oscillations decreases, and the oscillations themselves soon stop.

In order to reduce eddy induced currents in a pendulum oscillating in a magnetic field, its solid sector can be replaced with a comb with elongated teeth. Foucault currents will be reduced, and the pendulum will oscillate with virtually no damping.

As is generally accepted, “Foucault currents are currents that arise in a massive conductor located in an alternating magnetic field. Foucault currents have a vortex character. If ordinary induction currents move along a thin closed conductor, then eddy currents are closed inside the thickness of a massive conductor. Although at the same time they are no longer different from ordinary induction currents". According to Lenz's rule, these currents are directed in such a way as to counteract the cause that caused them. “Therefore, conductors moving in a strong magnetic field experience strong inhibition due to the interaction of Foucault currents with the magnetic field” . “Foucault currents shield the alternating magnetic field so that it does not penetrate deep into the conductor. However, Foucault currents cannot shield a static magnetic field, since due to ohmic resistance they cannot exist forever. The static magnetic field freely penetrates the conductor. However, the faster the field changes, the less depth it penetrates into the conductor. In good conductors, where ohmic losses are small, the decrease in field penetration depth becomes noticeable at very moderate frequencies.". It is believed that this is due to the demagnetizing effect of Foucault currents. It “it is more pronounced in the middle of the core and less on its surface, since areas in the middle of the core are covered by larger eddy currents than areas close to the surface”. As has been established, in superconductors this effect is inherent even in direct currents due to the lack of conductor resistance. “When a superconductor located in an external constant magnetic field is cooled, at the moment of transition to the superconducting state, the magnetic field is completely displaced from its volume. This distinguishes a superconductor from an ideal conductor, in which, when the resistance drops to zero, the magnetic field induction in the volume should remain unchanged.” .

Within theoretical physics, based on the general recognition of the vortex nature of Foucault currents, and therefore the vortex nature of the electric field, their description is based on an inductive pair of Maxwell’s equations:

Assuming that the density ρ is equal to zero free charges in Explorer and standard communication between current density and field strength

we obtain an equation for the magnetic field strength describing Foucault currents, as well as the skin effect:

At the same time "Eddy current strength according to Ohm's law equal to

where Φ m– magnetic flux coupled to the current circuit,R– resistance of the eddy current circuit. It is difficult to calculate this resistance. However, it is quite obvious that the smaller it is, the more conductivity conductor and the larger its dimensions" .

Therefore, to calculate losses from Foucault currents, approximate formulas are usually used, in which the specific losses depend on the type of iron, the thickness of the iron sheets, the frequency of the inducing field and the maximum induction of this field.

As we can see, the nature of Foucault currents is associated exclusively with the conductivity of the conductor and their structure is determined solely by the fact of the conductivity of metals, being the same for both ferro-, para-, and diamagnetic materials. The direction of these currents is opposite to the inducing one variable field, although these substances themselves behave fundamentally differently in external fields. As is known, diamagnets create their own field directed opposite to the external one, para- and ferromagnets create fields directed in the direction of the external magnetic field. Diamagnets include, in particular, inert gases, molecular hydrogen and nitrogen, zinc, copper, gold, bismuth, paraffin, etc., paramagnetic materials include aluminum; air. Ferromagnetic materials include, in particular, iron, nickel, and cobalt. But this difference is not considered to have a significant impact on the essence of Foucault's currents.

The experiments carried out also do not reveal this difference. Most of them come down to braking the fall of conducting bodies in a non-uniform magnetic field or to damping the oscillations of a metal pendulum. It is believed that for experiments “it is recommended to take copper or aluminum plates, since these materials have little resistivity. Consequently, the current strength in them will be greater and the effect will manifest itself more clearly" .

The second set of experiments with Foucault currents is associated with induction heating of both conducting bodies and dielectrics (in particular, drying wood). In theory this process the same foundation is laid, based on Maxwell's equations and the vortex nature of the inducing electric field. The use of a standard base also determines the emphasis on which the modeling is based. And although changes in the magnetic permeability of ferromagnets with temperature are taken into account, a significant difference in Foucault currents depending on the type of magnet is not made, as is limited to the case of a ferromagnet. In works devoted to the induction heating of aluminum, the phenomenological basis is also reduced to the standard representation of eddy currents exciting a field in the opposite direction to the exciting field, and the modeling of the process is based on this.

At the same time, for industrially produced household induction stoves, the main operating condition is the ferromagnetic material of the cookware used. With any other material, even non-ferromagnetic steel, the furnace refuses to work. This indicates certain nuances, which are not taken into account by the existing eddy current model, despite the abundance scientific developments and technological use of the process itself.

To study the characteristics of eddy currents, a special head with mutually perpendicular windings was developed, as shown in Fig. 1.

Rice. 1. Diagram and general view (a) of the head for studying eddy currents, as well as a diagram of instantaneous eddy currents in the core ( I 2) and in the cover 4 ( I 3) this head from the point of view of the standard concept (b) with instantaneous current in the primary winding I 1 ; 1 – core made of ferromagnetic material (transformer iron E330), 2 – primary single-row solid winding of 110 turns of wire ø0.23, 3 – secondary single-row solid winding of 110 turns of wire ø0.23, 4 – cover plate made of the material under study measuring 15x15x6 mm

Both head windings were wound on a movable fluoroplastic frame to adjust mutual perpendicularity. The size of the pad under study was chosen to be slightly larger than the space free from windings, for a purpose that will be clear from further research. Induction currents arising in the core and lining from the point of view of modern ideas about the counter-eddy nature of these currents are presented in Fig. 1b. As follows from this construction, when an asymmetric lining is applied, a current in the secondary winding cannot fundamentally arise due to the mutual perpendicularity of these currents to the turns of the secondary winding.

Electrical diagram The experiment is shown in Fig. 2.

Rice. 2. Electrical diagram of the experiment.

The experiment was carried out at a frequency of 20 kHz, the amplitude of the input signal was 2 V, the oscilloscope synchronization was external and was carried out according to the signal supplied to the primary winding of the head.

Four materials were used as linings, asymmetrically installed in the corners of the head: copper - diamagnetic, aluminum - paramagnetic, transformer iron and ferrite - ferromagnetic. The type of overlays is shown in Fig. 3.

Rice. 3. Type of overlays used in the study.

All overlays were made of several layers. The copper pad contained 8 layers, aluminum – 4 layers, iron – 20 layers and ferrite – 2 layers. All this was glued together with Stealth glue. The position indicators glued to each of the pads were set to their middle. The graduation scale on the head was also set to the middle of the primary winding, located vertically. General view installation is shown in Fig. 4.

Rice. 4. General view of the installation: 1 – oscilloscope, 2 – measuring head, 3 – signal generator, 4 – powerful output stage, 5 – power supply for the output stage

First of all, the very fact of induction in the secondary winding was investigated with the asymmetrical application of linings made of various materials. As already mentioned, synchronization was carried out using the input voltage to the primary winding of the head. The results of the experiment are presented in Fig. 5.

a) copper

b) aluminum

c) iron

d) ferrite

Rice. 5. Oscillograms induced emf in the secondary winding of the head (lower oscillogram) depending on the material and location of the lining on the head

As can be seen from the oscillograms, for copper and aluminum the induction emf is antiphase to the inducing current (right photos). The ferrite exhibits in-phase behavior in this position. Deviations for iron will be clarified further. In addition, it can be seen that moving the pad from the right corner to the left leads to a change in the phase of the emf by 180°. The difference in phases indicates that the nature of the occurrence of induced emf in ferromagnets, on the one hand, and para- and diamagnets, on the other hand, is different.

To reveal the trajectory of the induction emf, it was used that all the pads were made of plates. In this second experiment, the pads were placed in the same corner of the measuring head, along and across the plane of the head. The results are presented in Fig. 6.

a) copper

b) aluminum

c) iron

d) ferrite

Rice. 6. The nature of induction currents in pads made of the materials under study when they are rotated relative to the measuring head

From the oscillograms we see that when the copper and aluminum pads are rotated, the signal is significantly attenuated. This indicates that significant resistance to the eddy current arises. In ferrite, the signal almost does not change, which indicates the absence of an induction current, characteristic of copper and aluminum, but there is a current of the second type, characteristic of a ferromagnet. This current is in phase with the exciting one. In an iron plate, when turned to the end, not only the amplitude changes, increasing at the end when the Foucault currents decrease, but the phase of the signal also changes. This only happens when the resulting phase of the signal depends on the amplitudes of the original components, which is easy to show trigonometrically. Indeed, if we assume that the original components of the resulting signal are strictly shifted by approximately 180° and have different amplitudes, then

It is clear that when the amplitudes change due to changes in the conditions of current flow in the pads, the amplitude of the resulting signal will also shift AΞ, and the resulting phase φ Ξ. The described nature of the currents is presented in the construction shown in Fig. 7.

a) Induction currents in para- and diamagnets

b) Induction currents in ferrites

c) Induction currents in iron

Rice. 7. Electronic excitation circuit I e and orientation I c currents

In the case of para- and diamagnets, the end position of the pad (on the right) leads to the fact that instead of a single current I e it generates currents in each plate, which are induced not by the entire area of ​​​​contact of the pad with the inducing conductor, but only by a part limited by the thickness of the plate. This means that this inducing current, when the plate is rotated from the plane to the end, will also induce a smaller current in the secondary winding.

In the case of ferrite the situation changes. Current I c formed by molecular currents of ferrite. There is practically no electronic current in ferrite due to its high electrical resistance, and molecular currents depend little on the orientation of the ferrite, as a result of which rotation practically does not change the amplitude of the current in the secondary winding.

Both currents are present in the iron, and therefore the change in current I e leads as shown in general case us, to a change in both the amplitude and phase of the signal, since this current compensates for the current I c.

By the way, the competing action of these currents also leads to an incorrect physical interpretation of para- and diamagnetism, suggesting some special methods of rotating the orbitals of atoms in diamagnetic materials in order to create a field counter to the inducing one. As the above experiment showed, the difference between magnets comes down solely to the ratio of inducing currents. In diamagnetic I e exceeds I c, as a result of which an oncoming field is formed. In para- and ferromagnets, the ratio of currents is reversed, so a field is formed in the direction of the external inducing field. This feature also leads to incorrect measurements of the relative magnetic permeability of para- and diamagnetic materials. In fact, when the permeability of these substances is measured, it is measured with the compensating effect of the current I e. To measure real magnetic permeability, it is necessary to measure a finely dispersed phase of a substance held together by an insulating compound with μ = 1. This feature is also the cause of many paradoxes in electromagnetism.

You should also pay attention to the fact that the decrease in the induction current in the secondary winding is due to a decrease in the contact area of ​​the lining plate with the inducing conductor. Again, as in our previous experiments, it turns out that the inducing currents are excited not by some mythical magnetic field, but by a specific change mutual position conductors or by changing the current in the inducing conductor and for electron current I e proportional to the area of ​​contact of the conductor with the pad material. In fact, non-eddy currents are formed in the pad. The current arises exclusively in the contact area, and then it is closed through the body of the pad in the area of ​​weak inducing interaction. As a result, the electrical current circuit can be represented as shown in Fig. 8.

Rice. 8. Equivalent diagram of Foucault currents in para- and diamagnetic materials

According to this scheme, the electric field induced in para- and diamagnetic materials is not vortex. It remains potential, as in all other manifestations, but the current itself, excited in the material, is closed through the body of the conductor, creating the illusion of circularity.

The above is confirmed by the following two experiments. In the first of them, the opposite direction of the electron current is established I e and orientation molecular current I c. As we could see in the first of the above experiments, when the pad was shifted from one corner of the measuring head to another, the phase of the emf in the secondary winding always changed by 180° (or close to it). What happens if we install different materials on both corners of the head? In Fig. 9 shows the results of this operation. The pictures on the left show the emf in the secondary winding when installing one of the pads. In the pictures on the right - both of the overlays indicated in the caption to the pictures.

a) copper and aluminum

b) Iron (flat) and ferrite

c) Iron (end) and ferrite

d) Ferrite and copper

e) ferrite and aluminum

Eddy currents (Foucault currents)

Induction current occurs not only in linear conductors, but also in massive solid conductors placed in an alternating magnetic field. These currents turn out to be closed in the thickness of the conductor and are therefore called - vortex. They are also called Foucault's currents- named after the first researcher.

Foucault currents, like induced currents in linear conductors, obey Lenz's rule: their magnetic field is directed so as to counteract the change in magnetic flux that induces eddy currents. For example, if between the poles of an unswitched electromagnet a massive copper pendulum makes almost undamped oscillations, then when the current is turned on, it experiences strong braking and stops very quickly. This is explained by the fact that the resulting Foucault currents have such a direction that the forces acting on them from the magnetic field inhibit the movement of the pendulum. This fact is used to calm (dampen) the moving parts of various devices. If radial cuts are made in the described pendulum, then the eddy currents are weakened and braking is almost absent.

Eddy currents, in addition to braking (usually an undesirable effect), cause heating of the conductors. Therefore, to reduce heating losses, the armatures of generators and the cores of transformers are not made solid, but are made of thin plates separated from one another by layers of insulator, and they are installed so that the eddy currents are directed across the plates. Joule heat generated by Foucault currents is used in induction metallurgical furnaces. An induction furnace is a crucible placed inside a coil through which a high frequency current is passed. Intense eddy currents arise in the metal, which can heat it to the point of melting.

This method allows metals to be melted in a vacuum, resulting in ultra-pure materials.

Eddy currents also occur in wires carrying alternating current. The direction of these currents can be determined using Land's rule. In Fig. 182, A shows the direction of eddy currents as the primary current in the conductor increases, and in Fig. 182, b - when it decreases. In both cases, the direction of the eddy currents is such that they counteract the change in the primary current inside the conductor and promote its change near the surface. Thus, due to the occurrence of eddy currents, the fast-alternating current turns out to be unevenly distributed over the cross-section of the wire - it is, as it were, forced out onto the surface of the conductor. This phenomenon was called skin effect(from the English skin - skin) or surface effect. Since high-frequency currents practically flow in a thin surface layer, the wires for them are made hollow.

If solid conductors are heated with high-frequency currents, then as a result of the skin effect, only their surface layer is heated. The method of surface hardening of metals is based on this. By changing the field frequency, it allows hardening to be carried out at any required depth.

§ 126. Loop inductance. Self-induction

Electric current, flowing in a closed loop, creates a magnetic field around itself, the induction of which, according to the Biot-Savart-Laplace law, is proportional to the current. The magnetic flux F associated with the circuit is therefore proportional to the current I in the circuit:

where the proportionality coefficient L is called circuit inductance.

When the current in the circuit changes, the magnetic flux associated with it will also change; therefore, an emf will be induced in the circuit. Emergence of e.m.f. induction in a conducting circuit when the current strength changes in it is called self-induction.

From expression (126.1) the unit of inductance henry (H) is determined: 1 H is the inductance of such a circuit, the magnetic self-induction flux of which at a current of 1 A is equal to 1 Wb:

1 Hn=1 Vb/A=1 Vs/A.

It can be shown that the inductance of a circuit in general depends only on geometric shape contour, its dimensions and the magnetic permeability of the environment in which it is located. In this sense, the inductance of the circuit is analogous to the electrical capacitance of a solitary conductor, which also depends only on the shape of the conductor , its size and dielectric constant environment.

Applying Faraday's law to the phenomenon of self-induction (see (123.2)), we obtain that e. d.s. self-induction

If the contour is not deformed and the magnetic permeability of the medium does not change, then L=const and

. (126.3)

where the minus sign, due to Lenz's rule, shows that the presence of inductance in the circuit leads to slowing down change current in it.

If the current increases over time, then > 0 and < 0,t. i.e. the self-induction current is directed towards the current caused by external source, and slows down its growth. If the current decreases with time, then<0and > 0, i.e. the induction current has the same direction as the decreasing current in the circuit, and slows down its decrease. Thus, the circuit, having a certain inductance, acquires electrical inertia, which consists in the fact that any change in current is inhibited the more strongly, the greater the inductance of the circuit.

§ 127. Currents when opening and closing a circuit

With any change in current strength in a conducting circuit, an e occurs. d.s. self-induction, as a result of which additional currents appear in the circuit, called extra currents of self-induction. Extra currents of self-induction, according to Lenz's rule, are always directed so as to prevent changes in the current in the circuit, that is, they are directed opposite to the current created by the source. When the current source is turned off, the extra currents have the same direction as the weakening current. Consequently, the presence of inductance in the circuit slows down the disappearance or establishment of current in the circuit.

Let us consider the process of turning off the current in a circuit containing a current source with an emf. , resistance resistor R and an inductor L . Under the influence of external e. d . With. direct current flows in the circuit

At time t=0 we turn off the current source. The current in the inductor L will begin to decrease, which will lead to the appearance of an emf. self-induction, which, according to Lenz’s rule, prevents a decrease in current. At each moment of time, the current in the circuit is determined by Ohm's law, or

Dividing the variables in expression (127.1), we get . Integrating this equation over I (from I o to I) and t (from 0 to t), we find

where t=L/R is a constant called relaxation time. From (127.2) it follows that t is the time during which the current decreases by e times.

Thus, in the process of turning off the current source, the current strength decreases according to exponential law(127.2) and is determined by the curve 1 in Fig. The greater the inductance of the circuit and the lower its resistance, the greater t and, therefore, the slower the current in the circuit decreases when it opens.

When the circuit is closed, in addition to the external e. d.s . arises e. d.s. self-induction

preventing, according to Lenz's rule, an increase in current. According to Ohm's law,

By introducing a new variable , Let's transform this equation to the form

where t is the relaxation time.

At the moment of closure (t=0) current strength I=0 and u= - . Therefore, integrating over u (from - to IR - ) and t (from 0 to t ), we find

,

, (127.3)

Where - steady current (at t®¥).

Thus, during the process of turning on the current source, the increase in current strength in the circuit is given by function (127.3) and is determined by curve 2 in Fig. The current strength increases from initial value I=0 and asymptotically tends to the steady-state value. The rate of current increase is determined by the same relaxation time t= L/R as the current decrease. The establishment of current occurs the faster, the lower the inductance of the circuit and the greater its resistance.

Let's estimate the value of emf. self-induction arising with an instantaneous increase in the resistance of the DC circuit from R o to R. Let us assume that we open the circuit when a steady current I o flows in it = . When the circuit is opened, the current changes according to formula (127.2). Substituting the expression for I o and t into it, we get

E.m.f. self-induction

i.e., with a significant increase in the resistance of the circuit (R/R o >>1) with high inductance, emf. self-induction can be many times higher than the emf. current source included in the circuit. Thus, it is necessary to take into account that a circuit containing inductance cannot be abruptly opened, since this (the occurrence of significant self-induction emf) can lead to insulation breakdown and failure of measuring instruments. If resistance is gradually introduced into the circuit, then the emf. self-induction will not reach large values.

§ 128. Mutual induction

Let's consider two fixed contours (1 and 2), located quite close to each other (Fig. 184). If in the circuit 1 current flows I 1 , then the magnetic flux created by this current (the field creating this flux is shown in the figure as solid lines) is proportional to I 1 . Let us denote by Ф 21 that part of the flow that penetrates circuit 2. Then

where L 21 - proportionality factor.

If the current I 1 changes, then an emf is induced in circuit 2. , which, according to Faraday’s law (see (123.2)), is equal and opposite in sign to the rate of change of magnetic flux Ф 21 created by the current in the first circuit and penetrating the second:

.

Similarly, when current I 2 flows in circuit 2, the magnetic flux (its field is shown in Fig. 184 by dashed lines) penetrates the first circuit. If Ф 12 is part of this flow penetrating circuit 1, then

If the current I 2 changes, then in the circuit 1 induced by e.m.f. . , which is equal and opposite in sign to the rate of change of magnetic flux Ф 12 created by the current in the second circuit and penetrating the first:

.

The phenomenon of emf occurrence in one of the circuits when the current strength changes in the other is called mutual induction. The proportionality coefficients L 21 and L 12 are called mutual inductance of the circuits. Calculations, confirmed by experience, show that L 21 and L 12 are equal to each other, i.e.

. (128.2)

Coefficients L 12 and L 21 depend on the geometric shape, dimensions, relative position contours and from the magnetic permeability of the environment surrounding the contours. The units of mutual inductance are the same as for inductance , - Henry(Gn).

Let's calculate the mutual inductance of two coils wound on a common toroidal core. This case has great practical significance(Fig. 185). Magnetic induction of the field created by the first coil with the number of turns N 1, current I 1 and magnetic permeability m of the core, according to (119.2),

where l - core length midline. Magnetic flux through one turn of the second coil .

Then the total magnetic flux (flux linkage) through the secondary winding containing N 2 turns is

The flow y is created by the current I 1 therefore, according to (128.1), we obtain

(128.3)

If we calculate the magnetic flux created by coil 2 through coil 1, then for L 12 we obtain an expression in accordance with formula (128.3). Thus, the mutual inductance of two coils wound on a common toroidal core ,

.

Transformers

The operating principle of transformers used to increase or decrease alternating current voltage is based on the phenomenon of mutual induction. Transformers were first designed and put into practice by the Russian electrical engineer P. N. Yablochkov (1847-1894) and the Russian physicist I. F. Usagin (1855-1919). Schematic diagram transformer is shown in Fig. 186. The primary and secondary coils (windings), having N 1 and N 2 turns, respectively, are mounted on a closed iron core. Since the ends of the primary winding are connected to an alternating voltage source with emf. , then an alternating current I 1 arises in it, creating an alternating magnetic flux F in the transformer core, which is almost completely localized in the iron core and, therefore, almost completely penetrates the turns of the secondary winding. A change in this flux causes the appearance of an emf in the secondary winding. mutual induction, and in the primary - emf. self-induction. The current I 1 of the primary winding is determined according to Ohm's law:

,

where R 1 is the resistance of the primary winding. The voltage drop I 1 R 1 across resistance R 1 in rapidly varying fields is small compared to each of the two emfs, therefore

E.m.f. mutual induction arising in the secondary winding ,

. (129.2)

Comparing expressions (129.1) and (129.2), we find that the emf arising in the secondary winding, where the minus sign indicates that the emf. in the primary and secondary windings are opposite in phase.

The ratio of the number of turns N 2 /N 1 shows how many times the emf. In the secondary winding of the transformer there is more (or less) than in the primary winding, called the transformation ratio.

Neglecting energy losses, which in modern transformers do not exceed 2% and are associated mainly with the release of Joule heat in the windings and the appearance of eddy currents, and applying the law of conservation of energy, we can write that the current powers in both windings of the transformer are almost the same :

from where, taking into account relation (129.3), we find .

That is, the currents in the windings are inversely proportional to the number of turns in these windings.

If N 2 /N 1 > 1, then we are dealing with a step-up transformer that increases the variable emf. and reducing current (used, for example, to transmit electricity to long distances, since in in this case Joule heat losses, proportional to the square of the current, are reduced); if N 2 /N 1 < 1, then we are dealing with a step-down transformer that reduces the emf. and increasing current (used, for example, in electric welding, since it requires high current at low voltage).

We considered transformers with only two windings. However, transformers used in radio devices have 4-5 windings with different operating voltages. A transformer consisting of one winding is called an autotransformer. In the case of a step-up autotransformer, the emf. is supplied to part of the winding, and the secondary emf. is removed from the entire winding. In a step-down autotransformer, the mains voltage is supplied to the entire winding, and the secondary emf. is removed from part of the winding.

Induction currents can also be excited in solid massive conductors. In this case, they are called Foucault currents or eddy currents. The electrical resistance of a massive conductor is low, so Foucault currents can reach very high strengths.

In accordance with Lenz's rule, Foucault currents choose such paths and directions within the conductor so that their action can resist the cause that causes them as strongly as possible. Therefore, good conductors moving in a strong magnetic field experience strong inhibition due to the interaction of Foucault currents with the magnetic field. This is used to dampen (calm) the moving parts of galvanometers, seismographs and other instruments. A conductive (for example, aluminum) plate in the form of a sector is fixed on the moving part of the device (Fig. 63.1), which is inserted into the gap between the poles of a strong permanent magnet. When the plate moves, Foucault currents arise in it, causing inhibition of the system. The advantage of such a device is that braking occurs only when the plate moves and disappears when the plate is stationary.

Therefore, the electromagnetic damper does not at all interfere with the precise arrival of the system to the equilibrium position.

The thermal effect of Foucault currents is used in induction furnaces. Such a furnace is a coil powered by a high-frequency current of high strength. If you place a conducting body inside the coil, intense eddy currents will arise in it, which can heat the body to the point of melting. In this way, metals are melted in a vacuum, which makes it possible to obtain materials of exceptionally high purity.

Foucault currents are also used to heat the internal metal parts of vacuum installations for their degassing.

In many cases, Foucault currents are undesirable, and special measures have to be taken to combat them. So, for example, in order to prevent energy losses due to heating of transformer cores by Foucault currents, these cores are assembled from thin plates separated by insulating layers. The plates are arranged so that the possible directions of the Foucault currents are perpendicular to them. The appearance of ferrites (semiconductor magnetic materials with great electrical resistance) made it possible to manufacture solid cores.

Foucault currents arising in wires; through which alternating currents flow are directed in such a way that they weaken the current inside the wire and strengthen it near the surface. As a result, the fast-alternating current is distributed unevenly over the cross-section of the wire - it is, as it were, forced out onto the surface of the conductor. This phenomenon is called the skin effect (from the English skin - skin) or surface effect. The skin effect renders the interior of conductors in high-frequency circuits useless. Therefore, in high-frequency circuits, conductors in the form of tubes are used.