RU2522191C2 - Helmholtz-ishkov coils - Google Patents

Abstract

FIELD: electricity.

SUBSTANCE: invention is related to electric engineering and may be used both in laboratory conditions and in processing equipment using homogeneous magnet fields of different levels. The technical result consists in increase of the magnetic field homogeneity. Helmholtz coils are placed to an external shell-type magnetic core made of the upper and lower identical cylindrical cavities for the Helmholtz coils and surveyed objects coupled by their butt ends, and axes of these cavities are compliant. Outer faces of the magnetic core are rounded with a radius equal to the radius of the Helmholtz coils. In result of the magnetic resistance reduction in a space surrounding the Helmholtz coils the level of magnetic flux density is increased in a workspace of the device. At that deformation of the magnetic field lines takes place so that the lines in the central near-axial part of the workspace are reduced relatively to the larger extent than the lines in the peripheral part of the workspace.

EFFECT: invention contributes to improvement of the magnetic field homogeneity in the device workspace.

6 dwg, 7 tbl

Description

The present invention relates to electrical engineering and can be used to create devices with a uniform magnetic field, the length of which is comparable to the radius of the exciting windings. The technical result consists in increasing the uniformity of the magnetic field and the efficiency of its creation. The device consists of Helmholtz coils and an external magnetic circuit. Helmholtz coils typical round. The external magnetic circuit consists of two parts of a cylindrical shape with cylindrical recesses for placement of Helmholtz coils and blind end parts. An external magnetic circuit reduces the magnetic resistance of the external part of the magnetic circuit, thereby increasing the level or uniformity of the magnetic field between its bands.

The present invention relates to the field of magnetic field technology and can be used to create reference magnetic fields of a given level of uniformity and magnitude.

An analogue of the present invention are the Helmholtz coils themselves, which are a system of two identical round coils, which are placed coaxially at a distance of the radius of the coils and are turned on so that uniform magnetic lines of the magnetic field are formed and the magnitude of the magnetic field on the axis of the coil system between the planes of the coils is quite uniform .

The disadvantage of the analogue is the limited uniformity of the field and the effectiveness of its creation. To improve the uniformity of the magnetic field in the working area, additional pairs of large coils with an opposite inclusion of an additional field are also used. This increases the dimensions of laboratory equipment and reduces the energy efficiency of food, which must be stable.

The prototype of the invention is Ishkov's solenoid according to patent RU 2364000. It consists of a rectangular excitation coil and a spring magnetic circuit in the form of a cylindrical shell and two end flanges, the inner surfaces of which are poles. Inside the solenoid between the poles, when an electric current is passed through the excitation winding, a magnetic field of increased uniformity is created.

The disadvantage of the prototype in the light of the goal of the invention is that access to the internal space of the solenoid is associated with the need to dismantle the flanges with axial access to the working space of the solenoid. Particularly complex is the problem of radial access to the working space of the solenoid.

The technical solution to the problem of free access to the region of a uniform magnetic field along the axis and radius is a magnetic circuit of two cylindrical halves, of which the lower is fixed, and the upper is a removable part so that when the upper part is raised, access to the working space of the Helmholtz coils and along the axis, and along the radius. As an example, the device is shown in figure 1.

The device consists of Helmholtz coils 1 and an external cylindrical magnetic wire 2, which itself consists of the upper part 3 and the lower part 4. The ends of the magnet 2 are fixed along the radius of the Helmholtz coils 1. At the periphery of the implicit poles 5, the windings of the Helmholtz coils 1 are fixed 1. The object under investigation 6 fits in the central part of the device. Structurally, the magnetic circuit consists of two identical cylindrical parts with cylindrical recesses for Helmholtz coils 1, which are joined by flat ends. The difference between the upper lifting part 3 is the presence of a conical skirt 7, which mates with a conical cut 8 on the lower stationary part of the magnetic circuit 4.

The lifting of the upper part 3 and its fixation to the lower part 4 is carried out by known methods.

The device operates as follows.

When the upper part 3 is raised, the studied object 6 is placed in the central part of the device, connecting communications with it are carried out through the lower part 4 by known methods. When lowering the upper part 3, it is uniquely fixed to the lower part 4 by means of a conical skirt 7 and a conical cut 8.

When electric current is passed through the windings of Helmholtz coils 1 between the poles 5, the necessary magnetic field 9 occurs. The magnitude of the magnetic field is determined by the magnetomotive force of the coils, and the uniformity of the field is ensured by a specific redistribution of the structure of the magnetic field lines in the external magnetic circuit 2.

Figure 2 presents the course of the lines of the magnetic field in the present invention. The longest magnetic field line runs in the central part of the poles and has a length of 2πR + 2R (π + 1).

The shortest field line runs along the periphery of the poles and has a length of 2R, where R is the radius of the Helmholtz coils.

≃ $$\frac{1}{8}$$

, For the center line of the magnetic field, the ratio of the lengths of the interpolar part to the magnetic circuit is $$\raisebox{1ex}{$\mathrm{one}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.(\pi +\mathrm{one})$$

≃ $$\frac{\mathrm{one}}{8}$$

,

for the peripheral line of the magnetic field, this ratio will be

. $$\frac{R}{2R}=\frac{\mathrm{one}}{2}>\frac{\mathrm{one}}{8}$$

.

From the above relations it follows that for central magnetic lines only an eighth part passes between the poles, and for peripheral magnetic lines only half. Therefore, the use of an external magnetic circuit for Helmholtz coils deforms the lines of the magnetic field so as to reduce the magnetic resistance. For central magnetic lines, many times, but for peripheral magnetic lines - only twice. This will significantly increase the uniformity of the magnetic field in the present invention and increase its level at the same current in the Helmholtz coils.

Table 1 shows the radial dependence of the ratio of the length of the magnetic line in the magnetic circuit to the length of the magnetic line between the poles of the magnetic circuit l _F , which is determined by the above method. For simplicity, R = 1 is accepted, and r is the radial extension from the center of the poles. This is a nonlinear relationship.

Table 1 r 0 0.2 0.4 0.6 0.8 1,0

$$\frac{{\ell }_F=2\pi \left(\mathrm{one}-r\right)+\mathrm{one}}{{\ell }_o=\mathrm{one}}$$

7.28 6,024 4,768 3,512 2,256 1,0

The analytical calculation of the magnetic fields created by a thin conductor with current is based on the law of Bio-Savard-Laplace, see V. Savelyev. “Course in General Physics”, vol. 2, Moscow, Nauka, 1970, p. 132.

$$d\overrightarrow{B}=\frac{{\mu }_o}{\mathrm{four}\pi }\frac{id\ell }{r^2}\sin \left(d\overrightarrow{\ell }\widehat{}\stackrel{\rightharpoonup }{r}\right)$$

where µ _about = 4 · 10 ^-7 _, GN / m is the magnetic constant;

i is the electric current strength, A;

dl — elementary region with current I;

_- радиус-вектор из исследуемой точки к элементарному току; $$\stackrel{\rightharpoonup }{r}$$

_- radius vector from the studied point to the elementary current;

- угол между элементарным током $$d\overrightarrow{\ell }$$

и радиус-вектором $$\stackrel{\rightharpoonup }{r}$$

_, фиг.3. $$d\overrightarrow{\ell }\widehat{}\stackrel{\rightharpoonup }{r}$$

- angle between elementary current $$d\overrightarrow{\ell }$$

and radius vector $$\stackrel{\rightharpoonup }{r}$$

_, 3.

The differential of the magnetic field at the center of the circular current on its axis,

Where $$d\overrightarrow{B}=\frac{{\mu }_o}{\mathrm{four}\pi }\cdot \frac{id\ell }{R^2}$$

 R is the radius of the circular current, m

The magnitude of the magnetic field at the center of the circular current is determined by the integral $$\mathrm{AT}=\frac{{\mu }_o}{\mathrm{four}\pi }\frac{i}{R^2}{\displaystyle \int d\ell =}\frac{{\mu }_o}{\mathrm{four}\pi }\cdot \frac{i}{R^2}{\displaystyle \underset{o}{\overset{2\pi }{\int }}d\phi =\frac{{\mu }_o}{2}\frac{i}{R^2}}\left[Tl\right],$$

where dl = Rdφ.

Outside the plane of the coil, the magnetic field has two components, Fig.4. The axial component of the magnetic field depends on the angle β.

.B _z = Bsinβ, where sinβ = $$\beta =\frac{R}{\sqrt{R^2+z^2}}$$

.

The final form is the formula

$$B_Z=\frac{{\mu }_o}{2}\frac{R^2}{{\left(R^2+Z^2\right)}^{3/2}}$$

формула упрощаетсяwhere Z is the axial distance of the investigated point from the center of the circular current on its axis. For case $$R=\dot{z}=\mathrm{one}$$

the formula is simplified

$$B_z^{\text{'}\text{'}}=\frac{{\mu }_o}{2}\frac{\mathrm{one}}{{\left(\mathrm{one}+Z^2\right)}^{3/2}}.$$

For the second Helmholtz coil, this formula will take the form

$$B_Z^{\text{'}\text{'}\text{'}\text{'}}=\frac{{\mu }_o}{2}\frac{\mathrm{one}}{{\left(\mathrm{one}+{\left(\mathrm{one}-Z\right)}^2\right)}^{3/2}}$$

and generally on the axis of the Helmholtz coils will be

$$B_Z=B_z^{\text{'}\text{'}}+B_Z^{\text{'}\text{'}\text{'}\text{'}}=\frac{{\mu }_o}{2}\left(\frac{\mathrm{one}}{{\left(\mathrm{one}+z^2\right)}^{3/2}}\frac{\mathrm{one}}{{\left(\mathrm{one}+{\left(\mathrm{one}-Z\right)}^2\right)}^{3/2}}\right)$$

Calculations according to the above formulas are presented in table 2.

An analysis of the tabular data shows that the axial distribution of the level of magnetic induction appears to be a symmetric curvilinear dependence with a progressive decrease with distance from the center of the circular current. The same dependence for Helmholtz coils has a maximum in the center of the system of coils with a gentle slope when moving away from it. The field at the center of the coils exceeds the field at the center of the coils by 5.3%.

The absolute level of the magnetic field, even in multi-turn coils, is small, because the factor representing the magnetic constant is of the order of 10 ^-7 . For fields with a level of ITL = 10 ⁴ G, coils with the number of ampere-turns, measured in thousands, are required. Therefore, the use of an external magnetic circuit significantly increases the power supply efficiency of the coils and reduces the energy consumption for powering and cooling electromagnetic installations.

The radial distribution of the magnetic field in the plane of a circular coil with current can be investigated according to the scheme in figure 5,

where a is the radial displacement of the investigated point along the diameter of the circular current,

, произвольного, r is the distance of the studied point from the current element $$d\overrightarrow{\ell }$$

arbitrary

R is the radius of the circular current,

.φ is the azimuth of the current element $$d\overrightarrow{\ell }$$

.

For an arbitrary point lying on the diameter of a circular turn, we can write

$$\{\begin{array}{l}d\overrightarrow{B}=\frac{{\mu }_o}{\mathrm{four}\pi }\frac{id\ell }{r^2}\sin \left(d\overrightarrow{\ell }\widehat{}\stackrel{\rightharpoonup }{r}\right),\\ r^2=R^2+{\mathrm{but}}^2-2Ra\cos \phi \end{array}$$

Indeed, h = a · sinφ, b = a · cosφ.

Therefore, r ² = h ² + (Rb) ² = R ² + a ² -2Racosφ.

и вектором $$\overrightarrow{r}$$

$$d\overrightarrow{\ell }\widehat{}\stackrel{\rightharpoonup }{r}=\frac{\pi }{2}+\alpha ,$$

то $$\alpha =\mathrm{arc}sin\frac{asin\phi }{r}.$$

Since the angle between the current element $$d\overrightarrow{\ell }$$

and vector $$\overrightarrow{r}$$

$$d\overrightarrow{\ell }\widehat{}\stackrel{\rightharpoonup }{r}=\frac{\pi }{2}+\alpha ,$$

then $$\alpha =\mathrm{arc}sin\frac{asin\phi }{r}.$$

As a result of substituting the obtained expression into the initial formula of the BSL law and subsequent integration, we obtain the law of the radial distribution of magnetic induction over the diameter of the circular current

$$B\left(a\right)=\frac{{\mathrm{<figure-callout\; id="0"\; label="\mu "\; filenames="00000037.png,00000039.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_0}{2\pi }iR{\displaystyle \underset{o}{\overset{\pi }{\int }}\frac{\mathrm{one}}{r^2}sin\left(\frac{\pi }{2}+arcsin\frac{asin\varphi }{r}\right)}d\varphi$$

For the case R = i = 1, the integral is simplified

$$\{\begin{array}{l}B\left(a\right)=\frac{{\mathrm{<figure-callout\; id="0"\; label="\mu "\; filenames="00000037.png,00000039.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_0}{2\pi }{\displaystyle \underset{o}{\overset{\pi }{\int }}\frac{\mathrm{one}}{r^2}sin\left(\frac{\pi }{2}+arcsin\frac{asin\varphi }{r}\right)}d\varphi \\ r^2=\mathrm{one}`+a^2-2acos\varphi \end{array}$$

In the table. Figure 3 presents the results of numerical integration and experimental study of the radial distribution of magnetic induction in the plane of a circular coil with current.

In the author’s experiment, a round coil with a diameter of 100 mm was used from a copper wire with a diameter of 1 mm at a current of 35 A. At currents of a higher value, the coil was heated red-hot. The field was measured by a measuring coil with a diameter of 5 mm and a millivoltmeter. The last column shows the results of a study of a superconducting short solenoid with a diameter of 1.8 m, see Hawkins S.R. "Superconducting solenoids", ed. Mir, 1965, p.238-258.

The analysis of tabular data shows that the author’s theoretical calculation is correct and experimentally confirmed with sufficient accuracy.

The main conclusion is that the radial distribution of magnetic induction in the plane of a circular coil with a current is inhomogeneous and its heterogeneity grows progressively with distance to the periphery of the coil, where it increases many times. This value decreases axially, but increases radially in the plane of the coil and a conditional maximum of the saddle type falls at the center of the coil.

The magnitude of the magnetic induction outside the plane of the circular current can be determined if the test point is placed on a plane aligned with the axis of symmetry of the circular current, FIG. 6. The coordinates of this point will be: a, z, φ,

where a is the distance of the studied point from the axis of symmetry of the circular current,

z is the distance of the investigated point from the plane of the circular current,

φ is the azimuthal angle.

,Now the axial component of magnetic induction will be determined by the angle β according to the formula $$B_z\left(a,z\right)=B\left(a,z\right)sin\beta =B\left(a,z\right)\frac{r}{\rho }$$

,

where r ² = R ² + a ² -2Racosφ,

ρ ² = R ² + a ² + z ² -2Racosφ.

For R = i = 1, the calculation formula takes the form

$$\{\begin{array}{l}{B}_z\left(a,z\right)=\frac{{\mathrm{<figure-callout\; id="0"\; label="\mu "\; filenames="00000037.png,00000039.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_0}{2\pi }{\displaystyle \underset{o}{\overset{\pi }{\int }}\frac{r}{{\left(r^2+z^2\right)}^{3/2}}sin\left(\frac{\pi }{2}+arcsin\frac{asin\varphi }{r}\right)}d\varphi .\\ r=\mathrm{one}`+a^2-2acos\varphi \end{array}$$

In the table. 4 presents the results of the calculation according to this integral formula. The horizontal values are the relative values of magnetic induction during the radial translation of the calculated point, and vertically, respectively, with its axial movement. The value of magnetic induction in the center of a circular current is taken as 1.

Analysis of the contents of the table. Figure 4 shows that when the calculated point moves only along the symmetry axis of the circular current or only along the radius in the circular current plane, the numerical values of magnetic induction are repeated in accordance with Tables 2 and 3. Outside the circular current plane, the radial movement of the calculated point decreases monotonically.

In the table. 5 in relative units shows the distribution of magnetic induction in the diametrical plane of Helmholtz coils. An analysis of the contents of this table shows that this field is symmetric about the z = 0.5 plane. In the geometric center of the Helmholtz coils, the field has a conditional maximum of the saddle type, from which the ridge lines extend to the centers of the cross-section of the current-conducting conductors, when approaching which, the calculated value of the magnetic induction increases many times.

The inhomogeneity of the magnetic field at the center of the Helmholtz coils depends on the size of the selected region. For a cylinder with a height and diameter of 0.5 R, the heterogeneity is 1.5%.

In the case of coils of finite rectangular cross section, the BSL law takes the form $$dB=\frac{\mu o}{\mathrm{four}\pi }jdzdr\frac{dl}{r^2}\sin \left(d\overrightarrow{\ell }\widehat{}\stackrel{\rightharpoonup }{r}\right)$$

where jdzdr is the current element in the coil.

The magnitude of the magnetic induction at a point spaced a on the axis of symmetry of the coil and z on the median plane of the coil is determined by the triple integral

$$B\left(a,z\right)=\frac{\mu o}{2\pi }j{\displaystyle \underset{R_{\mathrm{one}}}{\overset{R_2}{\int }}{\displaystyle \underset{-\frac{Z_{\mathrm{one}}}{2}}{\overset{\frac{Z_{\mathrm{one}}}{2}}{\int }}{\displaystyle \underset{o}{\overset{\pi }{\int }}\frac{r_{\mathrm{one}}}{{\left(r_{\mathrm{one}}^2+z^2\right)}^{\frac{3}{2}}}\sin \left(\frac{\pi }{2}+\arcsin \frac{a\sin \varphi }{r_{\mathrm{one}}}\right)d\varphi dzdr}}}$$

Where $$r_{\mathrm{one}}^2=r^2+a^2-2ar\cos \varphi$$

R ₁ is the inner radius of the coil,

R ₂ is the outer radius of the coil,

- половина толщины катушки. $$\frac{Z_{\mathrm{one}}}{2}$$

- half the thickness of the coil.

In the table. 6 shows the results of calculations for R ₁ = 1, R ₂ = 1, 2, j = 30.

Analysis of the contents of the table. 6 shows that in the median plane of a coil of finite cross section, the magnitude of magnetic induction also increases with distance from the center of the coil, but the steepness of growth is weaker than in a thin single-coil coil. The axial decrease in the magnitude of magnetic induction persists, but it is also weaker.

In the table. 7 shows the diametrical distribution of the magnitude of the magnetic induction of thick Helmholtz coils.

In the center of thick Helmholtz coils, magnetic induction still has a maximum, but it is 15% higher than in thin ones with the same magnetomotive force. The general nature of the variation in the distribution of magnetic induction is maintained. The inhomogeneity deteriorated significantly and for a cylinder with a height and diameter of 0.5 R was 3% in height and 9% in radius.

The following conclusions should be made as general conclusions.

Coils with a winding cross-sectional diameter of less than 0.1 coil diameters can be considered thin with sufficient engineering accuracy.

For the coils of the final section of the winding, a universal integral formula is obtained for calculating the magnitude of the magnetic induction in the inner space of the coil.

When creating a magnetic system with a uniform magnetic field, one should not get carried away by thick coils.

Govorkova V.A. "Electric and magnetic fields", Moscow, Energia, 1968 on pages 205-207 provides an approximate calculation of the magnetic field inside a circular current, on pages 207-210 gives a calculation of the axial distribution of the magnetic field of a thin solenoid.

Claims (1)

A device for creating a uniform magnetic field, containing Helmholtz coils and an external armored magnetic circuit, characterized in that said magnetic circuit is made of upper and lower identical butt-joined parts with cylindrical cavities for Helmholtz coils and objects under study whose axes are aligned, the outer ends of the magnetic circuit are rounded with a radius equal to the radius of the Helmholtz coils.

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