A single-layer inductor is a wire coiled into a spiral. To provide rigidity, the wire is usually wound around a cylindrical frame. Therefore, in Coil32, the dimensions of the frame and the diameter of the wire are taken as the initial parameters, because they are easier to measure practically. The calculation formulas, however, use the geometric parameters of the spiral itself. To avoid confusion, you can read more about these subtleties on this help page.

Single-layer coils have become widespread, especially for shortwave and mediumwave amateur and broadcast band designs. The main properties of single-layer coils are high quality factor, relatively small internal capacity, and ease of manufacture. Let's consider methods for calculating such a coil without a gap between turns - " turn to turn"...

Let's start with the fact that at the end of the 19th century, H.A. Lorentz derived a formula using elliptic integrals to calculate the solenoid. The difference between the Lorentz model and the Maxwell model was the fact that the turns of the solenoid were represented not by an infinitely thin circular wire, but by an infinitely thin spiral conductive tape with a width equal to the actual thickness of the wire, without a gap between the turns. The formula is highly accurate when calculating a real coil if the latter has a large number of turns and is wound turn to turn. In 1909, the Japanese physicist H. Nagaoka transformed the Lorentz formula and brought it to a form from which an important conclusion followed - The inductance of the solenoid depends solely on the shape and size of the coil. Nagaoka's formula is as follows:

• L s - coil inductance
• N- number of coil turns
• r- winding radius
• l- winding length
• k L- Nagaoka coefficient

The most important conclusion from the analysis of this formula was that the Nagaoka coefficient depended only on the l/D ratio, which was called form factor coils. The Nagaoka coefficient was calculated using elliptic integrals. We will not dwell on this formula in more detail, because... Coil32 does not use it in calculations. It is only worth noting that in the case of a long solenoid, the formula simplifies to the following form:

where S is the cross-sectional area of ​​the coil. This formula is of academic interest only and is not suitable for calculating real coils, because is valid only for infinitely long solenoids, which do not exist in nature.

A single layer coil can be calculated numerically using Maxwell's formula or Nagaoka's solenoid formula. However, modern empirical formulas provide very high calculation accuracy and are quite sufficient for practical purposes.

We will begin the review and selection of empirical formulas with the most famous formula of G. Wheeler. Typically, this formula is most often used in various programs, online calculators, reference books and articles devoted to inductance calculations.

In the original, this formula looks like this:

L = a 2 N 2 / (9 a + 10 b)

Where N - number of turns, and a And b - the radius and length of the coil winding, respectively. Dimensions in inches. By adapting this formula for the metric system (or rather, for the GHS) and changing the radius to the diameter, we get the following:

• L- coil inductance [µH];
• N- number of coil turns;
• D- winding diameter [cm];
• l- winding length [cm];

This is our most famous version of this formula. Previously, on the website of the St. Petersburg University of Telecommunications - sut.ru there was a rather informative resource - dvo.sut.ru, where you could find a lot of information about inductors, including this formula. This resource has now been unfortunately deleted. But we managed to discover a clone of this resource on qrz.ru, to which even the old error migrated (0.5е1.0) in formula 2.37. There you can find both the Nagaoka formula (formula 2.28) and the expression for the Nagaoka coefficient through the Wheeler formula (formula 2.29).

The formula was proposed by Wheeler back in 1928, when computers were still only dreamed of and was very useful at that time, because made it possible to calculate a practical coil “in a column” on a piece of paper. The formula has become “rooted” in the mass consciousness of radio amateurs. However, few people know that it, like any empirical formula, has limitations. This formula gives an error of up to 1% for l/D > 0.4, that is, if the coil is not too short. This formula is not suitable for short coils.

Several attempts followed to eliminate this shortcoming. In 1985, R. Lundin published his two empirical formulas, one for “long” coils, the other for “short” coils, allowing one to calculate the Nagaoka coefficient with an accuracy of no less than 3ppM (±0.0003%), which is undoubtedly higher than the manufacturing accuracy or coil inductance measurements. Here is a calculator based on these formulas.
In 1982, 54 years later, with the advent of the computer era, Wheeler published his “long” formula, which calculated a single-layer coil with an error of no more than ±0.1%, both long and short. This formula was later improved by R. Rosenbaum, and subsequently by R. Weaver (Robert Weaver - analysis and derivation of the formula on his website).

• Dk- winding diameter
• N- number of turns
• k = l/Dk- coil form factor, the ratio of the winding length to its diameter

As a result, we have a formula that allows us to calculate a single-layer coil with an accuracy of at least 18.5 ppM (in comparison with the Nagaoka formula), which is worse than using the Lundin formulas, but firstly, it is quite sufficient for practical calculations, and secondly, we have one simpler formula instead of two, calculating a single-layer coil regardless of its form factor.

The formula is used in the online single-layer coil calculator, older versions of Coil32, as well as in all versions of the program for Linux and in the J2ME application for mobile phones.

The main version of Coil32 for Windows, as well as starting with version 3.0 for Android, uses a more complex method for calculating a single-layer coil, taking into account the spiral shape of the turns and an arbitrary winding pitch.

In 1907, E. Rosa, comparing calculations using Maxwell's method and Lorentz's method, derived

Wire selection. First of all, you should approximately select the diameter of the PEL or some other brand of wire. Since the calculation is simple, it can be performed for wires of various cross-sections and choose the one that gives the best results in terms of magnetic field strength with minimal power consumed by the electromagnet.

Having chosen the diameter of the wire, it is necessary to calculate its cross-sectional area 5pr and the permissible current strength for it/, based on its minimum density value equal to 2 a/mm 2,

I = 2S ave. (16)

For PEL brand wires, these data are given in the reference book.

Determining the length of the wire in the electromagnet winding. The total length of the wire l pr will be equal to

where U is the power source voltage, V;

R - winding resistance, ohm;

S np - cross-sectional area of ​​the wire, m 2 ;

ρ - copper resistivity equal to 1.7 * 10 -8 ohm * m 2 / m;

I - permissible current strength, a.

Calculation of the depth of the recess in the core and the number of layers (rows) of wire that fits in it. Knowing the depth a [equation (15)] of the recess in the core of the electromagnet and subtracting the insulation thickness δ and from it, the active depth of the recess is found

a ak = a - δ u. (18)

This value allows you to calculate the number of layers of wire that fit in this space. Since each layer of wire must be covered with a layer of transformer or capacitor paper δ mi = 0.02 mm, then the thickness of each winding layer will be

d pr + δ bi = d pr + 0.02 mm.

The number of layers n sl of the wire can be obtained by dividing the active depth a ak of the core recess by the layer thickness, i.e.

(19)

Determination of the length of the average turn of the winding. To find the total number of turns of an electromagnet winding, you need to know the length of the middle turn. To do this, it is necessary to first calculate the radii of the smallest and largest turns of the winding. The radius of the smallest turn r will obviously be equal to the sum

r min = r s + δ and + r pr, (20)

where r c is the radius of the electromagnet core, equal to half the diameter d p, mm;

δ and is the thickness of the insulation layer between the core and the winding, mm;

r r - radius of the wire with insulation, equal to half the diameter d r, mm.

The radius of the largest turn r max will be equal to

Knowing the radii of the smallest and largest turns, it is not difficult to calculate the radius of the middle turn as the arithmetic mean

(22)

The length of the average turn l avg will be equal to

l avg = 2πr avg. (23)

Determination of the total number of turns of wire and their number in one layer. Dividing the length of the wire l pr, found earlier, by the length of the Middle turn l cр, we obtain the total number of turns w in the winding

(24)

The number of turns w sl of wire in one layer can be found by dividing the total number of turns w by the number of layers n sl

(25)

Determination of the height of the notch in the electromagnet core. This value h p is calculated using the equation

(26)

where w sl is the number of turns of wire in one layer;

d pp - diameter of wire with insulation, mm;

δ and is the thickness of the insulation between the pole piece and the winding, mm;

α is the winding leakage coefficient, which can practically be taken equal to 0.98-0.99 *.

* (For small coil sizes, coefficient α can be taken equal to 1.)

Determination of the magnetic field strength in the electromagnet gap. Above, the dimensions of the electromagnet core, the number of turns of wire in it, and the size of the gap between the pole pieces and the body of the device were determined. Now you should check the compliance of the dimensions and winding of the electromagnet with its magnetic properties. To do this, it is necessary to calculate the magnetic field strength in the gap using the equation of the total current law

Iw = H 0 l 0 + H c l c + H t l t,

where I is the current strength in the winding, a;

w is the number of turns of wire in the winding;

l 0 - gap size, m;

Н с - magnetic field strength in the core, a/m;

l c - the value of the center line of the core, equal to the length of the magnetic flux in it, m;

N t - magnetic field strength in the electromagnet housing, a/m;

l t - length of the magnetic flux in the device body, m.

The values ​​of H c l c and H t l t can be neglected, since they are small compared to the value of H 0 l 0. Then the equation of the total current law in a simplified form will have the form

Iw = H 0 l 0 ,

H 0 = Iw / l 0 (27)

Such magnetic systems usually have an efficiency factor E f in the range of 0.8-0.9, so the calculation of the electromagnetic system can be considered completed if the magnetic field strength, taking into account E f equal to 0.8, is not less than 130,000-150,000 vehicles, i.e.

H 0 E f = 150,000.

Determining the amount of wire needed to make the winding. The reference literature gives the weight, m 100 m insulated wires, expressed in grams. The total weight of the TPR wire required for the manufacture of the electromagnet winding is equal to

m pr = l pr m / 100 g (28)

Determination of the total length of the device. Previously, from equations (13) and (26), the height of the pole pieces l p and the height of the recess h p for the electromagnet winding were found. Using these values, you can determine the length of the core L c in millimeters

L c = 2l p +h p (29)

To this value should be added L to the cover and bottom with the device pipes and L B for the free passage of water flow and placement of the electromagnet casing in the housing. Thus, the total length of the device L will be expressed by the sum L = L C + L to + L B mm.

Example. Calculation of a device for magnetic water treatment on a ZIL-130 engine. The distance between the pipes located on the radiator to the pipe on the water pump is 0.24 m (240 mm). The approximate size of the entire device can be taken equal to 0.2 m (200 mm).

The cross-sectional area S of the annular gap should be taken equal to the cross-sectional area of ​​the hose with an internal ∅ 0.045 m (45 mm) 0.0016 m2 (1600 mm2).

Based on the free space available on the engine, the diameter of the body Dk of the device can be taken equal to 120 mm.

With case wall thickness 4 mm the internal diameter of the device body must be

dk = 120 - 8 = 112 mm.

The diameter of the electromagnet casing D k0 will be equal according to equation (11)

The diameter D p of the pole piece is [according to equation (12)]

D p = D k0 - 2 (δ k + δ u) = 104 - 4 = 100 mm.

The height of the pole piece l p will be [see. equation (13)]

l p = D p / 4 = 100 / 4 = 25 mm

Electromagnet core diameter d p according to equation (14)

Structurally, this value can be reduced to 40 mm, i.e. d p ​​= 40 mm. Depth a of the recess for the electromagnet winding according to equation (15)

d = D p - d p / 2 = 100 - 40 / 2 = 30 mm.

The gap size l 0 between the electromagnet body and the pole piece is equal to

l 0 = d K - D p = 112 - 100 = 12 mm (0.012 m).

Example. Calculation of the electromagnet winding. Let a PEL wire with a diameter of 0.9 mm and with cross-sectional area 0.6362*10 -6 mg (0.6362 mm 2). The permissible current strength for a given wire cross-section is equal to 1.27 a.

The resistance of the entire electromagnet winding is

R = U / I * 12 * / 1.27 = 9.45 ohms.

* (Power source voltage - battery.)

Wire length l pr for winding an electromagnet

l pr = RS pr / ρ = 9.45*0.6362*10 -6 / 1.7*10 -3 = 353 m.

The active excavation depth according to equation (18) is equal to

and ak = a - δ u = 30 - 1 = 29 mm.

The number of layers of wire n sl that fits in this space, according to equation (19), is equal to

p sl = a - δ and / d pr + 0.02 = 29 / 0.96 + 0.02 = 29.6 or 29 layers.

The radius r min of the smallest turn [according to equation (20)] is equal to

r min = r s + δ I + r pr = 20 + 1 + 0.48 = 21.48 mm.

The radius of the maximum turn [according to equation (21)] is equal to

r max = r min + (d pr + 0.02) (n sl - 1) = 21.48 + 0.98*29 = 49.9 mm.

The radius of the middle turn r avg [see equation (22)]

Calculation of a DC electromagnetic drive with a retractable armature 1. Drive design
The design of a DC electromagnetic drive (EMD) with a retractable armature is shown in Fig. 1.1.

Rice. 1.1. Design of DC EMF with pull-in armature.
The EMF consists of a cylindrical steel housing in which a conductive (usually copper) winding is placed, which is a cylindrical solenoid. The case is closed on both sides with steel lids. A steel insert is installed on one of the covers. A steel anchor is inserted into the hole of the other cover. There must be a working gap between the armature and the core. The size of the working gap determines the maximum armature stroke. When electric current is passed through the winding, the armature creates a traction force, trying to be pulled into the winding. To return the armature to its original position when the current is turned off, a spring can be used (not shown in the drawing).
2. Statement of the problem
It is necessary to calculate the dependence of the maximum traction force of the EMF on the armature stroke. In Fig. Figure 2.1 shows a drawing of an EMF with dimensions.

Rice. 2.1. EMF drawing.
Accepted designations:
R0 - radius of insertion (anchor);
H0 - insertion height;
R1 - inner radius of the solenoid;
R2 - outer radius of the solenoid (inner radius of the drive housing);
H - solenoid height;
l- packaging factor;
j is the current density in the winding;
Rd - outer radius of the drive housing;
Hd is the height of the drive housing;
Z - working gap;
X - movement of the anchor from the initial position;
U - drive supply voltage;
I is the current value in the winding wire;
F is the force developed by the drive armature.
3. Calculation of the permissible current density in the windings
The heat generation power and, accordingly, the temperature of the winding depend on the current density in the winding. This temperature should not exceed that permissible for this brand of wire. The temperature inside the winding and, accordingly, the permissible current density in the windings can be calculated using the finite element method. The permissible current density in the winding wires depends on the design of the EMF and for solenoids with a winding thickness (R2 - R1) of up to 20 - 30 mm it can reach 5 ... 8 A/mm2 during long-term operation in an air environment with temperatures up to 40 0C.
If the packing factor is taken to be 0.6, then with a current density in the winding wire of 5 A/mm2, the current density in the winding itself will be 5 * 0.6 = 3 A/mm2. In this case, the temperature rise of the winding over the ambient temperature will be no more than 60 0C, and the heat resistance of the winding wire insulation should be approximately 100 0C.
If the current density in the winding wire reaches 7.5 A/mm2 (current density in the winding wire is 7.5 A/mm2, current density in the winding itself is 4.5 A/mm2), then the maximum winding temperature exceeding the ambient temperature during long-term operation will not exceed 120 0C . When winding, it is necessary to use wire with insulation of appropriate heat resistance.
4. Calculation of the maximum traction force of the EMF
The distribution of the magnetic field and the resulting forces can be calculated using the finite element method. The distribution of the magnetic field in the EMF is shown in Fig. 4.1.

Rice. 4.1. Magnetic field distribution in EMF.
5. Calculation of the EMF winding
The EMF winding is a cylindrical solenoid. Its calculation can be done in different ways, for example, using the Coil program. For a given solenoid size and for a given power source voltage, it is necessary to select the diameter of the winding copper wire so that the current density in the wire itself is as close as possible to the value obtained when calculating the maximum permissible current density (for example, 5 A/mm2).
6. Calculation examples
Example 1. EMF parameters:
R0 = 5 mm
H0 = 5 mm
R1 = 6 mm
R2 = 15 mm
H = 40 mm
l = 0.6
j = 3 A/mm2
Rd = 20 mm
Hd = 50 mm
U = 12 V

Clearance Z, mm 10 9 8 7 6 5 4 3 2 1 Stroke X, mm 0 1 2 3 4 5 6 7 8 9 Force F, N 1.71 1.84 2.02 2.25 2.57 3.00 3.72 5.18 7.86 16.60

Rice. 6.1. Dependence of the force developed by the EMF on the armature stroke.
When powering EMF from a 12-volt source, the winding should be wound with copper wire with a diameter (without insulation) of 0.27 mm. If the packing factor is 0.6, then the number of turns will be 3770, the resistance will be 73 Ohms, and the inductance will be 92 mH. The current consumption from a source with an output voltage of 12 V will be 0.17 A, power dissipation is about 2 W.
Example 2. EMF parameters:
R0 = 5 mm
H0 = 5 mm
R1 = 6 mm
R2 = 13 mm
H = 36 mm
l = 0.6
j = 3 A/mm2 or 4.5 A/mm2
Rd = 15 mm
Hd = 40 mm
U = 24 V

Clearance Z, mm 5 4 3 2 1 Stroke X, mm 0 1 2 3 4 Force F, N for current density 3 A/mm2 1.44 1.79 2.47 4.10 10.23 Force F, N for current density 4.5 A/mm2 3.16 3.88 5.27 8.38 17.22

Rice. 6.2. Dependence of the force developed by the EMF on the armature stroke.
When powering EMF from a 24-volt source with an allowable current density in the winding of 3 A/mm2, the winding should be wound with copper wire with a diameter (without insulation) of 0.16 mm. If the packing factor is 0.6, then the number of turns will be 7520, the resistance will be 373 Ohms, and the inductance will be 345 mH. The current consumption from a source with an output voltage of 24 V will be 0.064 A, power dissipation is about 1.5 W.
When powering EMF from a 24-volt source with an allowable current density in the winding of 4.5 A/mm2, the winding should be wound with copper wire with a diameter (without insulation) of 0.24 mm. If the packing factor is 0.6, then the number of turns will be 3340, the resistance will be 74 Ohms, and the inductance will be 68 mH. The current consumption from a source with an output voltage of 24 V will be 0.33 A, power dissipation is about 8 W.
If there is a margin for the developed force, then the supply voltage can be reduced accordingly, which will facilitate the thermal operation of the drive winding.
For questions regarding the calculation of specific EMF designs, please contact the author (see sectionContact Information ).
Links:

1. Coil: A program for calculating the parameters and magnetic field of a cylindrical solenoid
2. Brebbia K. et al. Boundary element methods: Transl. from English / Brebbia C., Telles J., Vroubel L. - M.: Mir, 1987. - 524 pp., ill.
3. Hromadka II T., Lei Ch. Complex method of boundary elements in engineering problems: Per. from English - M.: Mir, 1990. - 303 p., ill.
4. Kazakov L. A. Electromagnetic devices of REA: Handbook. - M.: Radio and Communications, 1991. - 352 p.: ill.
5. Norrie D., Freese J. Introduction to the finite element method: Transl. from English - M.: Mir, 1981. - 304 p., ill.
6. Silvester P., Ferrari R. Finite element method for radio and electrical engineers: Transl. from English - M.: Mir, 1986. - 229 p., ill.

Glossary of terms:

• Drive unit- a device having a working element capable of mechanical movement in the presence of a counterforce.
• Packing factor (fill factor)- ratio of the volume of the conductor to the volume of the winding; with uniform winding, it is equal to the ratio of the total area of ​​the conductors in the cross-section of the winding (without taking into account insulation) to the cross-sectional area of ​​the winding.
• Cylindrical solenoid- a solenoid in the form of a cylinder with a central cylindrical hole (if any).
• Electromagnetic drive- drive based on an electromagnet.

One day, once again, leafing through a book that I found near a trash can, I noticed a simple, approximate calculation of electromagnets. The title page of the book is shown in photo 1.

In general, their calculation is a complex process, but for radio amateurs, the calculation given in this book is quite suitable. Electromagnets are used in many electrical devices. It is a coil of wire wound on an iron core, the shape of which can be different. The iron core is one part of the magnetic circuit, and the other part, with the help of which the path of the magnetic lines of force is closed, is the armature. The magnetic circuit is characterized by the magnitude of magnetic induction - B, which depends on the field strength and magnetic permeability of the material. That is why the cores of electromagnets are made of iron, which has high magnetic permeability. In turn, the power flux, denoted in formulas by the letter F, depends on the magnetic induction. F = B S - magnetic induction - B multiplied by the cross-sectional area of ​​the magnetic circuit - S. The power flow also depends on the so-called magnetomotive force (Em), which is determined the number of ampere turns per 1 cm of the path length of the power lines and can be expressed by the formula:
Ф = magnetomotive force (Em) magnetic resistance (Rm)
Here Em = 1.3 I N, where N is the number of turns of the coil, and I is the strength of the current flowing through the coil in amperes. Other component:
Rм = L/M S, where L is the average path length of the magnetic power lines, M is the magnetic permeability, and S is the cross section of the magnetic circuit. When designing electromagnets, it is highly desirable to obtain a large power flux. This can be achieved by reducing the magnetic resistance. To do this, you need to select a magnetic core with the shortest path length of the power lines and the largest cross section, and the material should be an iron material with high magnetic permeability. Another way of increasing the power flow by increasing the ampere turns is not acceptable, since in order to save wire and power, one should strive to reduce the ampere turns. Usually, calculations of electromagnets are made according to special schedules. To simplify the calculations, we will also use some conclusions from the graphs. Suppose you need to determine the ampere turns and power flux of a closed iron magnetic circuit, shown in Figure 1a and made of the lowest quality iron.

Looking at the graph (unfortunately, I didn’t find it in the appendix) of the magnetization of iron, it is easy to see that the most advantageous magnetic induction is in the range from 10,000 to 14,000 lines of force per 1 cm2, which corresponds to from 2 to 7 ampere turns per 1 cm. For winding coils with the smallest number of turns and more economical in terms of power supply, for calculations it is necessary to take exactly this value (10,000 power lines per 1 cm2 at 2 ampere turns per 1 cm of length). In this case, the calculation can be made as follows. So, with the length of the magnetic circuit L = L1 + L2 equal to 20 cm + 10 cm = 30 cm, 2 × 30 = 60 ampere turns will be required.
If we take the diameter D of the core (Fig. 1, c) equal to 2 cm, then its area will be equal to: S = 3.14xD2/4 = 3.14 cm2. Here the excited magnetic flux will be equal to: Ф = B x S = 10000 x 3.14 = 31400 lines of force. The lifting force of the electromagnet (P) can also be approximately calculated. P = B2 S/25 1000000 = 12.4 kg. For a two-pole magnet this result should be doubled. Therefore, P = 24.8 kg = 25 kg. When determining the lifting force, it must be remembered that it depends not only on the length of the magnetic circuit, but also on the area of ​​​​contact between the armature and the core. Therefore, the armature must fit exactly against the pole pieces, otherwise even the slightest air gaps will cause a strong reduction in lift. Next, the electromagnet coil is calculated. In our example, a lifting force of 25 kg is provided by 60 ampere turns. Let us consider by what means the product N J = 60 ampere turns can be obtained.
Obviously, this can be achieved either by using a high current with a small number of coil turns, for example 2 A and 30 turns, or by increasing the number of coil turns while reducing the current, for example 0.25 A and 240 turns. Thus, in order for the electromagnet to have a lifting force of 25 kg, 30 turns and 240 turns can be wound on its core, but at the same time change the value of the supply current. Of course, you can choose a different ratio. However, changing the current value within large limits is not always possible, since it will necessarily require changing the diameter of the wire used. Thus, during short-term operation (several minutes) for wires with a diameter of up to 1 mm, the permissible current density, at which the wire does not overheat, can be taken equal to 5 a/mm2. In our example, the wire should have the following cross-section: for a current of 2 a - 0.4 mm2, and for a current of 0.25 a - 0.05 mm2, the wire diameter will be 0.7 mm or 0.2 mm, respectively. Which of these wires should be wound? On the one hand, the choice of wire diameter can be determined by the available assortment of wire, on the other hand, by the capabilities of the power sources, both in terms of current and voltage. Indeed, two coils, one of which is made of thick wire of 0.7 mm and with a small number of turns - 30, and the other of which is made of wire of 0.2 mm and a number of turns of 240, will have sharply different resistance. Knowing the diameter of the wire and its length, you can easily determine the resistance. The length of the wire L is equal to the product of the total number of turns and the length of one of them (average): L = N x L1 where L1 is the length of one turn, equal to 3.14 x D. In our example, D = 2 cm, and L1 = 6, 3 cm. Therefore, for the first coil the length of the wire will be 30 x 6.3 = 190 cm, the resistance of the winding to direct current will be approximately equal to? 0.1 Ohm, and for the second - 240 x 6.3 = 1,512 cm, R? 8.7 Ohm. Using Ohm's law, it is easy to calculate the required voltage. So, to create a current of 2A in the windings, the required voltage is 0.2V, and for a current of 0.25A - 2.2V.
This is the elementary calculation of electromagnets. When designing electromagnets, it is necessary not only to make the indicated calculations, but also to be able to choose the material for the core, its shape, and think through the manufacturing technology. Satisfactory materials for making mug cores are bar iron (round and strip) and various. iron products: bolts, wire, nails, screws, etc. To avoid large losses on Foucault currents, cores for alternating current devices must be assembled from thin sheets of iron or wire isolated from each other. To make iron “soft,” it must be annealed. The correct choice of core shape is also of great importance. The most rational of them are ring and U-shaped. Some of the common cores are shown in Figure 1.

Solenoid

A solenoid is an inductance coil made in the form of an insulated conductor wound on a cylindrical frame through which an electric current flows. The solenoid is a system of circular currents of the same radius, having a common axis in accordance with Figure 3.2-a.

Figure 3.2 - Solenoid and its magnetic field

If you mentally cut the turns of the solenoid across, designate the direction of the current in them, as indicated above, and determine the direction of the magnetic induction lines according to the “gimlet rule”, then the magnetic field of the entire solenoid will have the form as shown in Figure 3.2-b.

On the axis of an infinitely long solenoid, on each unit of length of which n 0 turns are wound, the field strength is determined by the formula:

At the point where the magnetic lines enter the solenoid, a south pole is formed, and where they exit, a north pole is formed.

To determine the poles of the solenoid, they use the “gimlet rule”, applying it as follows: if you place the gimlet along the axis of the solenoid and rotate it in the direction of the current in the turns of the solenoid, then the translational movement of the gimlet will show the direction of the magnetic field in accordance with Figure 3.3.

Figure 3.3 - Application of the gimlet rule

A solenoid, inside of which there is a steel (iron) core in accordance with Figure 3.4, is called an electromagnet. The magnetic field of an electromagnet is stronger than that of a solenoid because a piece of steel inserted into the solenoid is magnetized and the resulting magnetic field is strengthened.

The poles of an electromagnet can be determined, just like those of a solenoid, using the “gimlet rule”.

Figure 3.4 - Solenoid poles

The magnetic flux of a solenoid (electromagnet) increases with the number of turns and current in it. The magnetizing force depends on the product of the current and the number of turns (number of ampere-turns).

If, for example, we take a solenoid whose winding carries a current of 5A, and the number of turns of which is 150, then the number of ampere-turns will be 5*150=750. The same magnetic flux will be obtained if you take 1500 turns and pass a current of 0.5 A through them, since 0.5 * 1500 = 750 ampere turns.

You can increase the magnetic flux of the solenoid in the following ways:

a) insert a steel core into the solenoid, turning it into an electromagnet;

b) increase the cross-section of the steel core of the electromagnet (since for a given current, magnetic field strength, and therefore magnetic induction, an increase in the cross-section leads to an increase in the magnetic flux);

c) reduce the air gap of the electromagnet (since when the path of magnetic lines through the air is reduced, the magnetic resistance decreases).

Solenoid inductance. The solenoid inductance is expressed as follows:

where V is the volume of the solenoid.

Without the use of magnetic material, the magnetic flux density B within the coil is virtually constant and equal to

B = ?0Ni/l (3.9)

N - number of turns;

l is the length of the coil.

Neglecting edge effects at the ends of the solenoid, we find that the flux linkage through the coil is equal to the flux density B multiplied by the cross-sectional area S and the number of turns N:

This implies a formula for the solenoid inductance equivalent to the previous two formulas

DC solenoid. If the length of the solenoid is much greater than its diameter and no magnetic material is used, then when current flows through the winding inside the coil, a magnetic field is created directed along the axis, which is uniform and for direct current is equal in magnitude

Where? 0 - magnetic permeability of vacuum;

n = N / l - number of turns per unit length;

I is the current in the winding.

When current flows, the solenoid stores energy equal to the work that must be done to establish the current current I. The magnitude of this energy is equal to

When the current changes in the solenoid, a self-induction emf occurs, the value of which is

AC solenoid. With alternating current, the solenoid creates an alternating magnetic field. If the solenoid is used as an electromagnet, then on alternating current the magnitude of the attractive force changes. In the case of an armature made of soft magnetic material, the direction of the attractive force does not change.

In the case of a magnetic armature, the direction of the force changes. On alternating current, the solenoid has a complex resistance, the active component of which is determined by the active resistance of the winding, and the reactive component is determined by the inductance of the winding.

Application of solenoids. DC solenoids are most often used as a linear power drive. Unlike conventional electromagnets, it provides a long stroke. The power characteristic depends on the structure of the magnetic system (core and housing) and can be close to linear. Solenoids drive scissors for cutting tickets and receipts in cash registers, lock tongues, valves in engines, hydraulic systems, etc.

AC solenoids are used as an inductor for induction heating in induction crucible furnaces.