Skin effect in plasma. Skin effect and its application. Practical use of the skin effect

Consider the propagation of an electromagnetic wave in a conducting medium. For this we use Maxwell's equations (45.9) and take the rotor from the second of them. Taking and using the first and fourth equations, as well as the vector identity and Ohm's law, we obtain the equation for magnetic field:

Hence follows the dispersion equation

Consider the evolution of the initial state of the field (with a given Solving (87.2) with respect to and, we obtain

At, the magnetic field decays with a characteristic time. In a medium with good conductivity, there are two characteristic decay times

Note that for a fast decay, a for a slow one.

In a similar way, one can obtain an equation for the electric field in a medium, which has the form

where is the density of free charges. If they are absent, then the electric field decays in the same way as the magnetic one. In the presence of charges, the electric field can be represented as, where Then Eq. (87.5) splits into two, and the expression for coincides with (87.1), since Eevr The formula for ot takes the form

since Equation (87.6) is equivalent to the previously considered equation of relaxation of charges in a medium (23.1), which is easy to verify by taking the divergence from its left side. Therefore, like charges, the potential component of the field always decays with the characteristic time (87.4).

Let us now consider another problem: an electromagnetic wave of a given frequency and is incident on the boundary of a conducting medium. What is the attenuation of a wave in space? It is determined by the imaginary part. to from (87.2):

where is the characteristic depth of penetration of the alternating electromagnetic field into the conductive medium, called the thickness of the skin layer (from the English skin - skin).

In poorly conductive environments

where has the usual form. In the opposite limiting case

and the phase velocity

For an industrial frequency of 50 Hz (km), the thickness of the skin layer in copper is cm, and in iron it is mm, cm / s. In the radio range mm; (for copper).

Let us now find the relation between the electric and magnetic fields of the damped wave.The easiest way is to obtain it from the first equation (45.9): or, since

Since for good conductors (copper), otherwise in the radio range, so that it comes about the attenuation of the magnetic field. This great value is due to the reflection of the wave from the surface of a good conductor (see § 72), in which the electric fields of the incident and reflected waves almost cancel each other out. Relation (87.10) thus determines the so-called Leontovich boundary conditions for wave reflection from a conductor with finite conductivity for the field components tangent to the surface.

Problem 1. Calculate the resistance of the conductor taking into account the skin effect. From Ohm's law we find the total current in the skin layer:

The real part of this expression determines the ohmic resistance of the conductor (per unit length and unit of transverse size): imaginary - its internal inductance:

Let us now calculate the energy losses in the conductor. To do this, we find the modulus of the Poynting vector on the surface of the conductor. First of all, we obtain an expression for the vector product of complex vectors: where is the angle between them, directed from the vector a to Representing we obtain Thus,

This expression has a very simple physical meaning: the energy flux is equal to the energy density in the conductor near its boundary, multiplied by the speed of the wave movement inside the conductor

The same result can be obtained by direct integration of the Joule losses inside the conductor:

The most common application of the skin effect is for shielding against an alternating magnetic field. The latter can be harmful both in itself and due to the associated vortex electric field, which creates various electrical inductions. Shielding is achieved by surrounding the equipment to be protected with a sufficiently thick conductive screen. A practical difficulty is associated with the fact that usually the screen cannot be completely closed. For example, various openings are required for supplying power to the equipment, observing it, etc. It is interesting to note that such screens weaken the field more than according to a simple exponential law (see problems 2, 3).

Problem 2. Find the screening coefficient of a cylindrical screen of radius whose wall thickness is much less than the skin layer. The magnetic field is parallel to the cylinder axis.

In view of the condition of the field inside the walls, and hence the current density, can be considered uniform. Then the current in the screen (per unit of its length) can be determined simply by Faraday's law:

where is the field inside the screen. The law of conservation of the circulation of the magnetic field gives where is the external field. For the screening coefficient, we obtain

Here, in addition to the small factor that arises when the exponent is expanded, a large factor appears. The same multiplier appears with a strong skin effect. Physical reason additional weakening of the field in the shielded space is associated with the fact that the "tail" of the flow in solid metal is distributed over a large area. As a result, the following simple estimate is obtained for the screening factor:

Another important application of the skin effect is the formation of a magnetic field of the desired configuration, which repeats the shape of the conducting surface with an accuracy of the thickness of the skin layer.

The skin effect leads to a peculiar interaction of the alternating current with the conducting wall (Fig. XII.5). Since the lines of force do not penetrate deep into the conductor, then with a sufficiently small thickness of the skin layer, the normal component of the magnetic field on the surface is close to zero. Therefore, the configuration of the magnetic

Rice. XII.5. Fields of a pulsed electron beam near a conducting surface.

the current field near a conducting flat wall is equivalent to the field of two currents of different directions. One of them is usually called a current image by analogy with an electrostatic charge image. Thus, the current is "repelled" from the conductive surface.

If the current is created by a beam of charged particles, then in addition to the interaction of the current with the wall, there is also the interaction of the charge, which leads to the attraction of the beam by the wall. The latter is always stronger, so that the result is an attraction to the wall equal to the unit of beam length (cf. (30.4))

If we compensate for the electric charge of the beam, then the resulting force will change direction; such a beam will be repelled from the wall (Fig. XII.6). An interesting method of focusing a beam in a metal tube, cleverly called FUKO focusing, is based on this phenomenon. Since the bundle is repelled by the pipe "from all sides", it moves steadily along the pipe axis. This focusing makes it possible to transport a sufficiently intense beam along a curved tube and, in particular, to keep it in an annular tube.

Rice. XII.6. Reflection of an electron beam from a metal plate.

The name of this self-focusing is due to the fact that the currents induced by an alternating field in a conductor are known as Foucault currents, after the name of the French scientist who first described this phenomenon.

Problem 3. Estimate the magnetic field near the center of a thin conducting disk of radius and thickness placed in a uniform alternating magnetic field if

The Foucault currents excited by the density in the disk create a field on its axis (see (28.4))

In turn, the current in the don-center ring with the disk,

Ring resistance, is the total field in the plane of the ring. We emphasize that here the inductance of the ring is taken into account, since the EMF of the induction is calculated through the sum of the external field and the field of Foucault currents (compare (48.4) and problem 2).

The system of equations cannot be solved analytically. For estimation, you can take where is the field in the center of the disc. Then

(compare problem 2 and the commentary to it).

Let us now consider the unsteady skin effect, when the dependence of the magnetic field on time at the boundary of the conductor is not harmonic. If, as before, we neglect the displacement currents in comparison with the conduction currents, then from (87.1) we arrive at a diffusion-type equation:

The heat conduction equation has the same form (see (87.37) below). Magnetic field diffusion coefficient

The simplest case of a stationary skin effect corresponds to an exponential increase in the external field. This dependence is obtained from a harmonic formal substitution: Then for the one-dimensional problem the solution of the diffusion equation (87.14) is immediately obtained from (87.9) the same

Effective skin thickness

does not depend on time, as in the stationary case. Solution (87.16) can be interpreted as the diffusion propagation of the magnetic field front deep into the conductor

with speed

The last inequality is the condition for the applicability of the diffusion approximation (87.14), i.e., neglect of the displacement currents. For example, for copper with a diffusion rate

Let us now consider a more complicated problem of a nonstationary skin effect with a fast ("instantaneous") switching on of a harmonic field:

The frequency of the field and the thickness of the stationary skin layer are assumed to be equal to unity. Fourier spectrum of the field (87.20)

contains low frequencies which will determine a much stronger field penetration into the conductor compared to the stationary skin effect at frequency. Neglecting the latter (cf. spectra (87.21) and (78.8)) and considering the characteristic frequency range (see below), we can write the solution in the form of a Fourier integral:

We have used here the expression for the stationary skin effect at the frequency of the Fourier harmonic ω in the form

It is easy to check that this expression is true for both

The integral (87.22) is calculated by replacing the variable: and reducing the exponent to a perfect square (cf. (85.6)). As a result, we get

where is the new variable. Since the external field (87.20) can be represented in the form of the expression

describes a nonstationary skin effect when an external field is turned on and exactly coincides with the result obtained by another method.

At a fixed depth, the function reaches its maximum value

at the instant of time Thus, the maximum field decreases with depth much more slowly than in the case of a stationary skin effect. Note that at a given moment in time, the field inside the conductor has a maximum at equal

In the adopted approximation, all the expressions obtained are valid only for (see 87.23). Therefore, solution (87.24) does not satisfy the boundary condition where it is also necessary to take into account the rejected stationary contribution to the skin effect, which corresponds to frequencies in full spectrum(78.8) external field (87.20).

At high frequencies, the current flowing through the conductor is unevenly distributed over its cross section. Under the influence of strong magnetic fields of alternating current, the current is "pushed" from the center of the conductor to its surface (skin effect). As a result, the current flows over a smaller cross-sectional area, which looks like a decrease in the diameter of the wire. The higher the frequency, the less the thickness of the surface layer (skin layer) through which the current flows, and the greater the resistance of the conductor to the current flowing. Skin depth is defined as the distance below the surface where the current density falls 1 / e from the value at the surface (e is the base of the natural logarithm).

To minimize losses arising from the skin effect, conductors of a special design are used, which consist of a large number of thin strands, isolated from one another. The cores are intertwined so that each one passes along the surface and anywhere in the cross-section along the entire length of the wire; this averages the impedance of each strand so that equal currents flow through them. In such a conductor, called a litz wire (German Litzen - strands and Draht - wire), current flows along the surface of each core, as a result, the working cross-sectional area of ​​the conductor increases significantly, and the resistance to high-frequency currents decreases.

As a rule, when designing devices requiring the use of litz wire, the values ​​of the operating frequency and current in the conductor are known in advance. Since the main advantage of litz wire is the reduction in AC resistance compared to a single-core wire of equivalent cross-section, the main parameter that is taken into account when choosing a design and wire cross-section is the operating frequency. Table 1 shows the relationship between the AC and DC resistances (H factor) versus X factor for a single insulated round conductor:

Table 1.

where: d - wire diameter, mm, f - frequency, MHz.

From Table 1 and other empirical information, Table 2 was obtained, which shows the recommended diameters of a single strand of an insulated strand of a stranded wire, depending on the operating frequency.

Table 2.

After choosing the core diameter, the ratio between the AC and DC resistances of the ideal litz wire, i.e. one in which each core sequentially "penetrates" each point of the cross-sectional area can be determined by the following formula:

where: H - coefficient from Tables 1 and 2,

G - correction factor for eddy currents, determined by the formula:

N - number of cores in the cable, d1 - core diameter, mm,

d0 - bundle diameter, mm, f - frequency, Hz,

K - constant, depending on the number of cores in the cable, is determined from the following table:

Table 3.

The DC resistance of a stranded cable depends on the following factors:

1.section of the conductor,

2.number of lived,

3. the coefficient of elongation of a single strand in comparison with a unit length of a bundle arising as a result of weaving of strands. Typical values ​​are considered to be 1.5% for each order of the operation of weaving the cores into a bundle and 2.5% for

each order of the operation of twisting the bundles into a cable.

The following formula allows you to determine the DC resistance of a litz wire of any design:

where: RS is the resistance of a single core, Ohm (see table 2), NB is the number of orders of the braiding operation,

NC is the number of orders of operation for twisting bundles into a cable, NS is the total number of cores in the cable.

Example 1. Let's calculate the resistance of a type 2 wire (see Fig. 2), consisting of 450 cores with a diameter of 0.079 mm at a frequency of 100 kHz. This wire is made by twisting five bundles (twisting the bundles into a cable of the first order), each of which, in turn, is obtained by twisting three bundles (weaving of the second order), formed from

30 veins with a diameter of 0.079 mm (weaving first order).

1. We define active resistance wires according to the formula (4):

R = 3780.5 * (1.015) 2 (1.025) 1 = 8.87 Ohm / km,

2. We calculate the ratio R AC using the formula (2):

The advantage of litz wire becomes apparent when compared with 1.67 mm round wire having an equivalent cross-sectional area. The active resistance of a single-core wire will be about 7.853 Ohm / km, however, at a frequency of 100 kHz, the ratio between AC and DC resistance increases to about 21.4; so the AC resistance is

Example 2. Let's calculate the resistance of a type 2 wire (see Fig. 2), consisting of 1260 cores with a diameter of 0.100 mm at a frequency of 66 kHz. This wire is formed of seven bundles (twisting the bundles into a cable of the first order), each of which, in turn, is obtained by twisting six bundles (weaving of the second order), formed from 30 cores with a diameter of 0.100 mm (weaving of the first order).

1. Determine the active resistance of the wire by the formula (4):

A single-core wire with a diameter of 3.55 mm has the same cross-sectional area, but it is obvious that with a skin depth of 0.257 mm, such a wire can be considered as a thin-walled cylinder with a wall thickness equal to the skin depth.

Based on materials from New England Wire

Penetrating into the depth of the conductor, the amplitude in electromagnetic waves gradually decreases. This is the skin effect, which is also called surface effect. For example, if a current with a high frequency flows through a conductor, then its distribution does not occur over the entire section, but mainly in the surface layers.

How the skin effect works

This action should be considered on the example of a relatively long cylindrical conductor, which is influenced by an alternating voltage that has a certain frequency with a change in time.

If we take a constant voltage, the frequency of which is zero, then in this case the distribution of the electric current will be over the entire cross section of the conductor. This is due to the fact that the DC voltage will be the same at each point of the conductor cross-section. The lines of force of the magnetic field created by the current are formed in the form of concentric circles, the center of which coincides with the axis of the conductor. Thus, the direct current is distributed over the cross section, regardless of the effect of the magnetic field.

In the case of an alternating current in a conductor, it changes over time with a simultaneous change in the magnetic field. When the magnetic field flux changes, an electromotive force appears. It is this EMF that displaces the electric one to the surface of the conductor using a magnetic field. At very high frequencies, all current will only flow through the thin layer of the outer part of the conductor.

Skin effect properties

The skin effect is associated not only with high-frequency currents that change over time. This is due to any temporary change in currents. The skin effect can be observed when the conductor is directly connected to constant voltage... It is at this moment that a large induction EMF appears, which compensates for the effect of an external electric field on the axis. The end of this process is noted during the uniform distribution of the current in the conductor over the entire section.

With a very rapid change in current, a special time is found, during which the current and the magnetic field penetrate into the depth of the conductor. This value is called skin-new time. At the same time, one should take into account the factor that with a decrease in the resistivity of the conductor, the time of penetration of current and magnetic field into it increases.

In the case of using superconductors, the skin-time, theoretically, will have an infinitely large value, no magnetic field is observed, and the current flows exclusively over the surface.

Skin effect

Skin effect (from the English skin - skin, shell), surface effect, weakening of electromagnetic waves as they penetrate deep into the conductive medium, as a result of this effect, for example, alternating current high frequency or alternating current over the cross section of the conductor or alternating magnetic flux over the cross section of the magnetic circuit, when flowing through the conductor, is not evenly distributed over the cross section, but mainly into the causes of the effect.

Reasons for the effect.

The skin effect is due to the fact that when an electromagnetic wave propagates in a conducting medium, eddy currents arise, as a result of which part of the electromagnetic energy is converted into heat. This leads to a decrease in the strengths of the electric and magnetic fields and the current density, i.e. to attenuation of the wave.

Eddy currents, Foucault currents, closed electric currents in a massive conductor, which arise when the magnetic flux penetrating it changes. Eddy currents are induction currents and are formed in a conducting body either due to a change in time of the magnetic field in which the body is located, or due to the movement of the body in a magnetic field, leading to a change in the magnetic flux through the body or any part of it. The value of the eddy current is the greater, the faster the magnetic flux changes. /

The higher the frequency n of the electromagnetic field and the greater the magnetic permeability m of the conductor, the stronger (in accordance with Maxwell's equations) the vortex electric field created by the alternating magnetic field, and the greater the conductivity a of the conductor, the greater the current density and the power dissipated per unit volume (in according to Ohm's and Joule's - Lenz's laws). Thus, the more n, m and s, the stronger the damping, i.e. The skin effect is more pronounced.

Maxwell's equations, fundamental equations of classical macroscopic electrodynamics, describing electromagnetic phenomena in an arbitrary medium. Maxwell's equations were formulated by J.K. Maxwell in the 60s of the 19th century on the basis of a generalization of the empirical laws of electrical and magnetic phenomena. Based on these laws and developing the fruitful idea of ​​M. Faraday that interactions between electrically charged bodies are carried out by means of an electromagnetic field, Maxwell created a theory of electromagnetic processes, mathematically expressed by Maxwell's equations.The modern form of Maxwell's equation was given by the German physicist G. Hertz and the English physicist O. Heaviside. Maxwell's equations connect the quantities characterizing the electromagnetic field with its sources, that is, with the distribution of electric charges and currents in space. In a vacuum, the electromagnetic field is characterized by two vector quantities that depend on spatial coordinates and time: the electric field strength E and the magnetic induction B. These quantities determine the forces acting from the field on charges and currents, the distribution of which in space is set by the charge density r (charge in unit volume) and current density j (charge transferred per unit time through a unit area perpendicular to the direction of movement of charges). To describe electromagnetic processes in a material medium (in a substance), in addition to vectors E and B, auxiliary vector quantities are introduced, depending on the state and properties of the medium: electric induction D and magnetic field strength H. Maxwell's equations allow determining the main characteristics of the field (E, B , D and H) at each point in space at any time, if the sources of the field j and r are known as functions of coordinates and time. Maxwell's equations can be written in integral or differential form (below they are given in the absolute system of units of Gauss; see CGS system of units). Maxwell's equations in integral form are determined by the given charges and currents not by the field vectors E, B, D, H themselves at separate points in space, but by some integral quantities depending on the distribution of these field characteristics: the circulation of the vectors E and H along arbitrary closed circuits and flows vectors D and B through arbitrary closed surfaces. Maxwell's first equation is a generalization to alternating fields of the empirical Ampere's law on the excitation of a magnetic field by electric currents. Maxwell hypothesized that the magnetic field is generated not only by currents flowing in conductors, but also by alternating electric fields in dielectrics or vacuum. The quantity proportional to the rate of change of the electric field in time was called by Maxwell the displacement current. The displacement current excites the magnetic field according to the same law as the conduction current (this was later confirmed experimentally). The total current, equal to the sum of the conduction current and the displacement current, is always closed.

First M. at. looks like:

/

In the case of a plane sinusoidal wave propagating along the x-axis in a well-conducting, homogeneous, linear medium (displacement currents compared to conduction currents can be neglected), the amplitudes of the electric and magnetic fields decay exponentially:

Damping coefficient, m0 - magnetic constant At the depth х = d = 1 / a, the wave amplitude decreases by a factor of e. This distance is called the penetration depth or skin thickness. For example, at a frequency of 50 Hz in copper (s = 580 ks / cm; m = 1) s = 9.4 mm, in steel (a = 100 ks / cm, (m = 1000) d = 0.74 mm. an increase in frequency to 0.5 MHz d will decrease 100 times. An electromagnetic wave does not penetrate at all into an ideal conductor (with infinitely high conductivity), it is completely reflected from it. The shorter the distance that the wave travels, in comparison with d, the weaker it appears S.-e.

Magnetic constant, the coefficient of proportionality m0, which appears in a number of formulas of magnetism when written in a rationalized form (in the International System of Units). So, the induction B of the magnetic field and its strength H are related in vacuum by the ratio

B = m0H,

where m0 = 4p × 10 -7 gn / m "1 .26 × 10 -6 gn / m.)).

For conductors with a strongly pronounced Skin effect, when the radius of curvature of the wire cross-section is much greater than d and the field in the conductor is a plane wave, the concept of the surface resistance of the conductor Zs (surface impedance) is introduced. It is defined as the ratio of the complex amplitude of the voltage drop per unit length of the conductor to the complex amplitude of the current flowing through the cross section of the skin layer of unit length.

Complex amplitude, representing the amplitude A and phase y of harmonic oscillation x = Acos (wt + y) using a complex number = Aexp (ij) = Acosj + iAsinj. In this case, the harmonic vibration is described by the expression

x = Re [( expiwt)],

where Re is the real part of a complex number in square brackets. K. a. usually used in the calculation of linear electrical circuits (with linear relationship current from voltages) containing active and reactive elements. If a harmonic emf of frequency w acts on such a circuit, then the use of K. and. current and voltage allows you to go from differential equations to algebraic. The connection between K. and. current I and voltage U for active resistance R is determined by Ohm's law: / = R. For inductance L, this relationship has the form I = - and for capacity C: I = iwCU. Thus, the quantities iwL and L / iwC play the role of inductive and capacitive resistance. /

Complex resistance per unit length of the conductor:

where R0 is the active resistance of the conductor, which determines the power losses in it, X0 is the inductive resistance, which takes into account the inductance of the conductor due to the magnetic flux inside the conductor, lc is the perimeter of the cross-section of the skin layer, w = 2pn; in this case, R0 = X0. With strongly expressed S. - e. surface resistance coincides with the characteristic impedance of the conductor and, therefore, is equal to the ratio of the electric field strength to the magnetic field strength on the surface of the conductor.

/! Characteristic impedance of transmitting electrical lines, the ratio of voltage to current at any point in the line along which electromagnetic waves propagate. V. s. is the resistance that a line has to a traveling voltage wave. In an infinitely long line or a line of finite length, but loaded on a resistance equal to V. s., There is no reflection of electromagnetic waves and the formation of standing waves. In this case, the line transfers almost all the energy from the generator to the load (without losses). V. s. equals:

/

In cases where the mean free path l of current carriers becomes greater than the thickness d of the skin layer (for example, in very pure metals at low temperatures), at relatively high frequencies the skin effect acquires a number of features, due to which it is called anomalous. Since the field along the electron mean free path is inhomogeneous, the current at a given point depends on the value of the electric field not only at this point, but also in its vicinity, which has dimensions of the order of l Therefore, when solving Maxwell's equations, instead of Ohm's law, one has to use the kinetic Boltzmann equation to calculate the current ... Electrons with the anomalous Skin effect become unequal in terms of their contribution to electricity; for l >> d, the main contribution is made by those of them that move in the skin layer parallel to the metal surface or at very small angles to it and spend, that is, more time in the strong field region (effective electrons). Attenuation of an electromagnetic wave in the surface layer still takes place, but quantitative characteristics the abnormal skin effect is somewhat different. The field in the skin layer does not decay exponentially (R0 / X0 = ).

In the infrared frequency range, the electron may not have time to cover the distance l during the period of the field change. In this case, the field on the path of the electron for the period can be considered uniform. This leads again to Ohm's Law, and the Skin effect becomes normal again. Thus, at low and very high frequencies, the skin effect is always normal. In the radio range, depending on the ratio between / and d, normal and abnormal skin effects can occur. All of the above is true as long as the frequency is less than the plasma one: w< w0 «(4pne2/m) 1/2 (n - концентрация свободных электронов, е - заряд, m - масса электрона).

Fighting the effect.

The skin effect is often undesirable. In wires, alternating current with a strong skin effect flows mainly along the surface layer; in this case, the cross-section of the wire is not fully used, the resistance of the wire and the power loss in it at a given current increase. In ferromagnetic plates or tapes of magnetic cores of transformers, electrical machines and other devices, an alternating magnetic flux with a strong skin effect passes mainly along their surface layer; as a result, the use of the cross-section of the magnetic circuit deteriorates, the magnetizing current and losses in the steel increase. The "harmful" effect of the Skin effect is weakened by a decrease in the thickness of the plates or tape, and at sufficiently high frequencies - by the use of magnetic cores made of magnetodielectrics.

Magnetodielectrics, magnetic materials, which are a mixture of a ferromagnetic powder and a binder - a dielectric (for example, Bakelite, polystyrene, rubber) bound into a single conglomerate; in macro volumes have a high electrical resistance depending on the amount and type of bundle. M. can be both magnetically hard materials and magnetically soft materials. Soft magnetic metals are produced mainly from fine powders of carbonyl iron, molybdenum permalloy, and alsifer with various bonds. Soft magnetic magnetic materials are used for the manufacture of cores for inductors, filters, chokes, and radio-technical armor cores operating at frequencies of 104-108 Hz. /

Also, with an increase in the frequency of the alternating current, the skin effect manifests itself more and more clearly, which forces it to be taken into account when designing and calculating electrical circuits working with alternating and impulse current. For example, copper wires coated with a thin layer of silver can be used instead of conventional copper wires. Silver has the highest conductivity among all metals, and its thin layer, in which, due to the skin effect, more ́ Most of the current has a strong effect on the resistance of the conductor. The skin effect significantly affects the characteristics of the oscillatory circuits, such as the quality factor. Due to the fact that high-frequency current flows through the thin surface layer of the conductor, the active resistance of the conductor increases significantly, which leads to a rapid damping of high-frequency oscillations. To combat the skin effect, conductors of various cross-sections are used: flat (in the form of ribbons), tubular (hollow inside), a layer of metal with a lower resistivity is applied to the surface of the conductor. For example, in HF equipment, silver-plated copper circuits are used, in high-voltage power lines, a wire in a copper or aluminum sheath with a steel core is used, in high-power alternators, the winding is made of tubes through which liquid hydrogen is passed for cooling. Also, in order to suppress the skin effect, a system of several twisted and insulated wires is used - litz wire. All of these methods of combating the skin effect are ineffective for microwave equipment. In this case, oscillatory circuits of a special shape are used: volumetric resonators and specific transmission lines Application of the effect

Applying an effect.

On the other hand, the Skin effect finds application in practice. The skin effect is based on the action of electromagnetic shields. So, to protect the external space from interference created by the field of a power transformer operating at a frequency of 50 Hz, a screen of relatively thick ferromagnetic steel is used; for shielding the inductor operating at high frequencies, the shields are made of a thin layer of Al. The skin effect is based on high-frequency surface hardening of steel products (see. Induction heating unit).

Induction heating installation, electrothermal installation for heating metal blanks or parts using induction heating. /

Also, the effect of explosive magnetic generators (EMG), explosive magnetic frequency generators (EMFG) and, in particular, shock-wave emitters (UVI) is based on the skin effect.

The depth of the conductor layer, in which the electric field strength decreases by a factor of e, is called the skin depth. The dependence of the skin depth on the frequency for a copper conductor is shown in the table. - waveguides. surface layer.

Formula for calculating skin depth in metal (approximate).

Here ε0 = 8.85419 * 10 -12 F / m - absolute dielectric constant of vacuum, ρ - resistivity, c - speed of light, μm - relative magnetic permeability (close to unity for para- and diamagnets - copper, silver, etc. ), ω = 2π * f. All quantities are expressed in SI units.

A simpler formula for calculating

ρ - resistivity, μm - relative magnetic permeability, f - frequency.

Everyone knows that a plasma ball does not shock. Although a voltage of tens of thousands of volts passes through a person ... Why ???

If a very high voltage is applied to the plasma ball - more than 100KV - the discharges will begin to leave the glass bulb. Again, these sparks can be “touched”, only you will not feel anything.

Let's remove the ball from the stand.

And finally, disconnect the stand itself from the Tesla coil.

In all 4 cases, a current of 100-200KV passes through a person, but why does it not have any effect? Is the current strength small? No, by including in the circuit> Tesla coil -> wire -> spark -> man< лампу накаливания (если в ней будет хотя бы один виток волоска - опыт не получится), можно заставить волосок нагреться.

The answer is simple: the high-frequency current passes only along the surface of the conductor (skin), causing only heating. But do not think that the discharge from the Tesla coil is completely safe for 2 reasons.

) some sparks may have a low frequency

) there will be a burn at the place where the spark enters the body.

To avoid burns, hold a small metal, NOT insulated object (such as a screwdriver, piece of foil or wire) in your hand.

During the experiments, a 450W Tesla coil was used, turned on at medium power to prevent damage WEB cameras who was filming.

SKIN system is a reliable and safe complex designed for heating pipelines of various lengths for underwater, underground and aboveground laying, as well as in areas with increased explosion hazard.

SKIN system is the only possible heating method for pipelines without an accompanying network, the length of which can be up to 30 thousand meters;

· the system is designed with high levels of reliability and durability;

· SKIN effect makes it possible to heat highways of any length;

· can be used in areas of increased explosion hazard;

· elements for heating have a heat release rate of up to 120 watts per meter;

· SKIN system works at temperatures up to 200 degrees;

· there is a permit for use in areas of increased explosion hazard from the Federal Service for Environmental, Technological and Nuclear Supervision and a certificate of compliance with GOST R;

· there is no potential on the outer parts of the elements that generate heat, they do not need electrical insulation, since they are grounded.

Appointment

SKIN system (Induction-Resistive System) allows you to maintain the specified temperatures of pipelines, protects them from freezing, makes it possible to heat up pipelines of any length.

The SKIN system is unique, since it alone can heat a pipeline leg with a line length of up to 30 thousand meters with power supply without a support network. The SKIN effect allows you to obtain economically advantageous heating of highways of any length in the presence of an escort network.

Operating principle

Tesla electromagnetic skin effect

Pipe and conductor currents are directed towards each other, which causes proximity and surface effects. The current in the pipe passes through the inner layer, and there is no voltage on its surface. The conductor is made of aluminum or copper (non-magnetic materials), so there is no significant surface effect, and the alternating current flows through the conductor section. The main element that generates heat in the SKIN system is the pipe, which takes up about 80 percent of the system power.

Advantages

Large length of the heated section of the pipeline.

A small system resistance per meter of length in combination with a high power supply voltage makes it possible to supply up to 30 thousand meters of heating arms.

Powering occurs from one end. In its essence, the design solution of the system allows powering the heating area from one end.

Electrical safety. The outer part of the heating element is at zero potential with respect to earth and is grounded.

Good thermal contact. The heating element (metal) is fixed (with special fasteners) or welded to the pipeline. To improve contact (thermal), a paste with good thermal conductivity is used.

Ease of installation. There is no external thermal insulation on the fuel elements, which makes it impossible to damage it during installation work.

Increased reliability. A pipe made of steel (low-carbon) guarantees protection of the conductor from various damage and mechanical strength, which is important for highways that are laid under water and ground.

Heat dissipation

The working temperature range is from -50 degrees to +200 degrees. Electricity supply ranges from 50 Hertz to 5 kilowatts.

Structural elements include:

The heat-generating element is a steel pipe with a diameter of 20-60 mm and a wall thickness of at least 3 mm.

Conductor. A special conductor is used as a current-carrying conductor, which withstands mechanical loads during installation work, thermal loads up to 200 degrees and high voltage up to 5 kW.

Corrosion protection - if required by the customer, an epoxy coating can be applied.

Control

To increase efficiency, the IRSN system is equipped with a special control device that reduces the heating power when the outside air temperature rises. Such a control device guarantees careful control over the system state and makes it possible to detect emergency conditions, which is important.

An example of heating a heat-insulated pipeline with three heating elements of the SKIN-system with a total power of 130 W / m.

Pipe diameter 530 mm, t env. Air = - 20 °

Power supply diagram of the pipeline section heated by the SKIN-system

Pipeline section with SKIN-system heating (power supply circuit). The electrical power system includes transformer substation complete type (KTP), with cells (distribution) of the low and high sides, a special transformer (balancing), a control and monitoring system. The complete transformer substation is installed in a heated sealed container.

Bibliography

1)Netushil A.V., Polivanov K.M., Fundamentals of Electrical Engineering, vol. 3, M., 1956;

2)Polivanov K.M., Theoretical basis electrical engineering, part 3 - Theory of the electromagnetic field, M., 1975;

)Neiman L.R., Surface effect in ferromagnetic bodies, L. - M., 1949.

)Kalashnikov S.G., Electricity, M., 1956 (General course of physics, vol. 2).

)Tolmasskiy I.S., Metals and alloys for magnetic cores, Moscow, 1971.

1. Surface effect ………………………………………………… ..2

2. Electrical surface effect on the example of a rectangular tire …………………………………………………… .3

3. Calculation of the complex impedance of the tire ............................................................. 9

4. Magnetic surface effect ……………………………………… 11

5. Calculation of the complex power in a sheet in a sinusoidal magnetic flux ………………………………… ... 15

6. Analysis of expressions for specific complex power …………… 17

7. Approximate methods of calculating the complex power in a steel sheet in a magnetic flux. …………………………………………………………………………………………………………… ..... 18

8. Electrical surface effect in a conductor with a circular cross-section ………………………………………………………… .21

9. The effect of proximity ……………………………………………………… ..26

10. Complex resistance of the tire in the presence of proximity effect ……………………………………………………………………………………………………………………………………………………………… 30

11. Parameters of single-phase busbar trunking ………………………………… 33

12. Electromagnetic fields and parameters of busbars of three-phase busbar trunking ……………………………………………………………… ..34

13. Calculation of the field in tires C, B, A …………………………………………… ... 36

14. Calculation of the complex impedance of the tire …………………………… 38

15. Equivalent equivalent circuits of a three-phase busbar trunking with a symmetrical system of currents ……………………………………… ... 40

16. Electromagnetic field in the cable sheath …………………………… .45

17. Complex resistance of the shell ………………………………… .47

18. References ……………………………………………………… ... 49

Surface effect

It has been experimentally established and theoretically confirmed that an alternating electric current (including sinusoidal), in contrast to a constant current, is unevenly distributed over the cross-section of the conductor. In this case, there is always a tendency for the current to be displaced from the inner part of the conductor to the peripheral part, i.e. the current density in a conductor increases as you move from depth to the surface of the wire. This phenomenon is called electrical surface effect. It can be explained as follows.

It was previously indicated that the Poynting vector has a component normal to the lateral surface of the conductor, and this indicates the penetration of energy into the conductor from the surrounding space through this surface. At the same time, it was noted that electromagnetic waves propagate in the direction of the Poynting vector and decay in the same direction in a conducting medium. But if this is so, then in a conductor streamlined by current, the current density, as well as the electric and magnetic strengths at the surface should be greater than in depth. Another more graphic explanation can be given to the electrical surface effect. If a current lead is flown around by a sinusoidal current, then its internal parts are coupled with a large magnetic flux compared to the peripheral ones, and therefore, in accordance with the law of electromagnetic induction, large electromotive forces will be induced in them, preventing the change in current and being practically in antiphase with the current density vector. For this reason, it can be assumed that in the inner parts of the conductor, the total electric strengths and current densities are interconnected by Ohm's law () , will have lower values ​​than in the peripheral.

If the current frequency and parameters are such that the wave penetration depth is much less than the conductor cross-section (Δ« d), then the current in the conductor will be concentrated only in a thin surface layer, the thickness of which is practically determined by the depth of wave penetration. This surface effect is called pronounced. The displacement of the current leads to an increase in the active resistance of the current lead in comparison with its value at constant current. It is for these reasons that in high-frequency installations the inductor is made in the form of a copper tube, inside which a liquid is passed for cooling.

If the depth of wave penetration is commensurate with the overall dimensions, then the conductor is called transparent and it is believed that the current is distributed almost evenly over the cross section of this conductor.

If an alternating magnetic flux is closed in a conducting ferromagnet, then it is also displaced onto the surface of the magnetic circuit, the magnetic induction and intensity increase in the surface layer, and this entails an increase in the eddy current density and Joule losses.

In the case of a magnetic surface effect, the penetration depth of the wave is also introduced into consideration, and provided that Δ« d, the effect is considered to be pronounced. The phenomenon of the magnetic surface effect is widely used in electrothermics, however, in electric machines, transformers and other similar installations, the manifestation of this effect is highly undesirable.

Electrical surface effect on the example of a rectangular tire

In fig. 1 shows a bus of rectangular cross-section, streamlined by current I. The field in the bus satisfies the Helmholtz equation

There is an electromagnetic field and a conduction current inside the bus. Outside the bus (conductivity (γ=0) conduction current (δ=0) absent, but electric and magnetic fields exist. Since the internal and external electromagnetic fields are interconnected, when solving the problem of calculating the field inside the tire, it is necessary to know the laws of the field distribution and outside it.

Thus, with a strict approach, you need solve the problem of calculating the field in the entire space - inside and outside the tire.

Since this problem is very difficult for an exact analytical solution, we will formulate conditions and assumptions under which the problem of the surface effect in a tire can be solved approximately with good accuracy. First, consider the field in a round wire (Fig. 2).

Magnetic lines are concentric circles. V this example the flux caused by the current in the wire is divided into two components - internal and external. This property of a round wire is used in engineering practice to determine the internal inductance of a wire. As seen from Fig. 3, with a square cross-section of the wire, such a clear delineation of flows cannot be made, since the cross-sectional contour is no longer a line of force.

Determine what impact the geometry of the tire has (h/2 a) on the distribution of the field in its volume. From fig. 4 it follows that as the relative sizes increase (h/ 2a) the lines of force inside the tire begin to take shape, approaching the shape of the outer contour of the tire. If the attitude h/2 a » 1 (Fig. 5), then practically in the entire volume of the tire, the vector of magnetic intensity becomes directed along the larger lateral surface of the tire, that is. towards coordinates at.

If we now neglect the edge effects, then for the bus at h»2a it is possible to solve the problem in the coordinate system (x, y,z) assuming that

,
,

,
.

Fig. 4 Fig. 5

NS let's leave the task: calculate field distribution E and N in the volume of a rectangular bus (Fig. PO) and calculate its complex resistance to a sinusoidal current if the bus h / 2a "1 flows around the current I with frequency ω .

Rice. 6 Fig. 7

Environment parameters: μ , γ ... Accepted assumption Ė=Ė x (z) leads to the Helmholtz equation (index NS in what follows we will omit) with respect to the vector of electrical intensity

, (5.34)

where
.

The solution to equation (5.34) is the set of exponential functions

, (5.35)

. (5-36)

Let us write down the general solution for , using Maxwell's second equation
. Since in the case under consideration
, then

. (5.37)

In view of (5.35)

. (5.38)

Next, we find the integration constants WITH 1 and WITH 2 ... Since the investigated field has the symmetry
, therefore, from (5.35) we have

Obviously, the last equality is valid if WITH 1 = C 2 = C / 2.

Then, taking into account the symmetry condition, expressions (5.35) and (5.38) will have the form, respectively

, (5.39)

. (5.40)

Integration constant WITH proportional to the current set in the bus I.

Let's select some area dS= hdz (fig. 7). Then

(5.41)

J n

From here we find
. (5.42)

As a result, the final solution for Ė looks like:

. (5.43)

Substitution of (5.42) into (5.40), taking into account (5.34), makes it possible to obtain a solution for the magnetic strength:

. (5.44)

Thus, (5.43) and (5.44) are the final expressed for the electric and magnetic strengths and into the volume of the tire.

Of interest is a qualitative analysis of the current density distribution in the bus volume (Fig. 8). According to Ohm's law
for the current density in the bus, we have

.

Distribution pattern δ(z) will obviously depend on the propagation coefficient
.

If on low frequencies parameter a / ∆ small (ra<< 1) , then for a small argument shpz≈1 , Shpa≈ pa and then

Thus, under these conditions, the current is evenly distributed over the bus and the surface effect does not appear. As the frequency increases, the picture changes, since with increasing parameter (ra) the unevenness of the current distribution over the cross-section of the bus increases.