RU2490659C1 - Method for nondestructive volume measurement of vector function of flux density of magnetic field nonuniformly distributed in space and periodically varying in time - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: in the method, measurement of instantaneous volume states of distribution of a magnetic field that is nonuniform in space is carried out at points that are not accessible by mechanical penetration. The working magnetic measuring tool is made in form of one loop oriented in space and tied to a spherical coordinate system. The working tool performs movement which involves alternating discrete parallel displacements in the direction of the axis of the vector of the normal of the tool and revolutions of the direction of said displacements, given by zenith and azimuthal angles of the spherical coordinate system, such that the discrete parallel displacements are repeated at different angles.

EFFECT: broader functional capabilities of magnetic measurement.

2 dwg

Description

The invention relates to information-measuring equipment, in particular to magnetometry, and can be used for non-destructive registration in places inaccessible to mechanical penetration of instantaneous volumetric states of the distribution of a magnetic field that is inhomogeneous in space and periodically varies in time. The registration result is the values of the vector induction function determined at points in space and time points, reconstructed by applying the inverse Radon transform to the measured magnetic flux projections, obtained by controlled spatial movement of the working body, integrating the vector field function along the plane.

A known method of topography of a magnetic field, implemented in the device [1], which is based on measuring the magnetic field simultaneously in a large number of plane points by means of a regular matrix structure of columns and rows of interconnected magnetosensitive elements located on a semiconductor wafer placed in the field. However, the known method and the device realizing it do not allow measurements to be made in places inaccessible to mechanical penetration of the semiconductor wafer and allow one to obtain only one component of the vector function of magnetic induction, equally directed with the normal to the semiconductor wafer. Also, it is not possible to perform measurements at arbitrary points, since all measurement points are determined by the matrix structure of magnetically sensitive elements. In addition, the method has limitations on the number of magnetically sensitive elements and, as a consequence, on resolution.

A known method of scanning a magnetic field, implemented in the device [2], which is based on measuring the magnetic field alternately at the points of the plane by means of a series-connected matrix of sensitive elements, including N flux probes, followed by processing the measured voltage using a computer. However, the known method and the device that implements it do not allow measurements in places inaccessible to mechanical penetration of the matrix of sensitive elements and allow you to get only one component of the vector function of magnetic induction. Also, it is not possible to perform measurements at arbitrary points, since all measurement points are determined by the matrix structure of magnetically sensitive elements. In addition, the method has limitations on the number of magnetically sensitive elements and, as a consequence, on resolution.

A known method of measuring and topography of magnetic scattering fields near the surface of an object, implemented in a device [3], which is based on sequential movement in accordance with a predetermined path using a measuring rod of one three-component Hall sensor relative to the measurement object by means of a mechanical movement unit with a rotary table and movable carriages driven by stepper motors, followed by statistical processing of the measurement results th block. However, the known method and the device implementing it do not allow measurements to be made in places inaccessible to mechanical penetration of the Hall sensor.

Closest to the claimed is a method of obtaining the distribution of the vector function of the magnetic induction of a periodic magnetic field [4], implemented in the device [5], based on successive translational and angular movements of the magnetically sensitive working body, while the volume under investigation is represented by a combination of parallel sections, the distribution of magnetic induction in which are obtained by applying the procedure of reconstruction of computational tomography to the voltages induced in accordance with Faraday in the hold circuits, moving in a magnetic field. However, the known method and the device implementing it do not allow measurements to be made in places inaccessible to mechanical penetration of the magnetically sensitive working body.

The technical result of the application of the proposed method is to expand the functionality of magnetometry, which consists in measuring the distribution of the vector function of the magnetic induction of a periodically varying field at certain points in the space under study for arbitrarily selected time instants over a period. The claimed method is based on sequentially translational movements and rotations through the corners of the working magnetically measuring organ and registration of the stresses induced in it, by applying the inverse Radon transform to the measured projections of the magnetic flux obtained using the working organ during its controlled spatial movement in the measurement volume by the parallel formation method source projection data.

The technical result implemented in the method of non-destructive volumetric measurement of the vector function of magnetic induction non-uniformly distributed in space and periodically changing in time of the magnetic field is achieved by measuring instantaneous volumetric states of the distribution of a non-uniform in space magnetic field in places inaccessible to mechanical penetration, and the worker the magnetic measuring organ is performed in the form of a space-oriented one contour tied to a spherical a coordinate system, moreover, in the course of its controlled spatial displacement, a method of parallel formation of the initial projection data of the induction function is performed, for which a working body performs a movement that involves alternating discrete parallel displacements in the direction of the axis of the normal vector of the organ and rotations of the directions of these displacements specified by the zenith and azimuthal angles of spherical coordinate systems, so that discrete parallel movements are repeated many times at different angles and, moreover, for the zenith in the range from 0 to π, and for the azimuthal in the range from 0 to π, the initial projection data of the Cartesian components of the distribution of the vector function of magnetic induction in the measurement volume in the Cartesian coordinate system, necessary for the reconstruction algorithm, are obtained by trigonometric transformations:

$$\{\begin{array}{l}{p}_x\left(s,\overline{n},t\right)=\left[sin\theta \cdot p\left(s,\overline{n},t\right)+cos\theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right)\right]cos\alpha -\\ -\left[sin\theta \cdot p\left(s,{\overline{n}}_{\perp \alpha },t\right)+cos\theta \cdot p\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)\right]sin\alpha ;\\ p_y\left(s,\overline{n},t\right)=\left[sin\theta \cdot p\left(s,\overline{n},t\right)+cos\theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right)\right]sin\alpha +\\ +\left[sin\theta \cdot p\left(s,{\overline{n}}_{\perp \alpha },t\right)+cos\theta \cdot p\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)\right]cos\alpha ;\\ p_z\left(s,\overline{n},t\right)=cos\theta \cdot p\left(s,\overline{n},t\right)-sin\theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right),\end{array}$$

- исходные проекционные данные функции индукции; s - координата оси направления управляемого пространственного перемещения рабочего органа; θ и α - зенитный и азимутальный углы отклонения рабочего органа, соответственно; $${\overline{n}}_{\perp \theta }=\left(n,\theta +\pi /2,\alpha \right)$$

, $${\overline{n}}_{\perp \alpha }=\left(n,\theta ,\alpha +\pi /2\right)$$

и $${\overline{n}}_{\perp \theta ,\alpha }=\left(n,\theta +\pi /2,\alpha +\pi /2\right)$$

- векторы нормалей плоскостей, соответственно перпендикулярно повернутых по углам θ и α на величину равную π/2 относительно текущего вектора нормали рабочего органа $$\overline{n}$$

; t - время; p_x, p_y, p_z - исходные проекционные данные x-, y-, z-компонент векторной функции магнитной индукции, которые записывают через магнитные потоки, численно равные интегралам по времени напряжений, индуцируемых в контуре:Where $$p\left(s,\overline{n},t\right)$$

- initial projection data of the induction function; s is the coordinate of the direction axis of the controlled spatial movement of the working body; θ and α are the zenith and azimuthal angles of deviation of the working body, respectively; $${\overline{n}}_{\perp \theta }=\left(n,\theta +\pi /2,\alpha \right)$$

, $${\overline{n}}_{\perp \alpha }=\left(n,\theta ,\alpha +\pi /2\right)$$

and $${\overline{n}}_{\perp \theta ,\alpha }=\left(n,\theta +\pi /2,\alpha +\pi /2\right)$$

are the normal vectors of the planes, respectively, perpendicularly rotated at angles θ and α by an amount equal to π / 2 relative to the current normal vector of the working body $$\overline{n}$$

; t is the time; p _x , p _y , p _z are the initial projection data of the x-, y-, z-components of the vector function of magnetic induction, which are written through magnetic fluxes, numerically equal to the time integrals of the stresses induced in the circuit:

$$\{\begin{array}{l}{p}_x\left(s,\overline{n},t\right)=-\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]cos\alpha +\\ +\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]sin\alpha ;\\ p_y\left(s,\overline{n},t\right)=-\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]sin\alpha -\\ -\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]cos\alpha ;\\ p_z\left(s,\overline{n},t\right)=-cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt,\end{array}$$

where u (t) are the stresses induced by a change in the flow in the circuit at time t; T is the period during which one complete change in the function of magnetic induction occurs; n = 1, 2, ... is the period number, and the inverse Radon transform based on their convolution filtering using the convolution function, which is the inverse Fourier transform of the square of the frequency of the spatial spectrum, is applied to the initial projection data:

$$\{\begin{array}{l}{B}_x\left(x,y,z,t\right)={\displaystyle \int \left(p_x\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_x\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]cos\theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}[{\displaystyle \underset{0}{\overset{\pi }{\int }}((-\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]cos\alpha +}}\\ +\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]sin\alpha )\ast h\left(s\right))d\alpha ]cos\theta d\theta ;\\ B_y\left(x,y,z,t\right)={\displaystyle \int \left(p_y\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_y\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]cos\theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}[{\displaystyle \underset{0}{\overset{\pi }{\int }}((-\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]sin\alpha +}}\\ +\left[sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]cos\alpha )\ast h\left(s\right))d\alpha ]cos\theta d\theta ;\\ B_z\left(x,y,z,t\right)={\displaystyle \int \left(p_z\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_z\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]cos\theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(-\left[cos\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt-sin\theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\ast h\left(s\right)\right)d\alpha }\right]cos\theta d\theta ,}\end{array}$$

- дифференциал вектора нормали; dθ и dα - дифференциалы зенитного и азимутального углов, соответственно; B_x(x,y,z,t₀), B_y(x,y,z,t₀), B_z(x,y,z,t₀) - x-, y-, z-компоненты векторной функции магнитной индукции $$\overline{B}$$

, соответственно, благодаря чему реконструируют декартовы компоненты распределения векторной функции магнитной индукции в пространстве.where the symbol "*" is a convolution operator; h (s) is a convolution function; $$d\overline{n}$$

- the differential of the normal vector; dθ and dα are the differentials of the zenith and azimuthal angles, respectively; B _x (x, y, z, t ₀ ), B _y (x, y, z, t ₀ ), B _z (x, y, z, t ₀ ) - x-, y-, z-components of the vector function magnetic induction $$\overline{B}$$

, accordingly, due to which the Cartesian components of the distribution of the vector function of magnetic induction in space are reconstructed.

The essence of the method of non-destructive volumetric measurement of the vector function of magnetic induction non-uniformly distributed in space and periodically changing in time of the magnetic field is to register in places inaccessible to mechanical penetration, instantaneous volumetric states of the magnetic field distribution, due to which the values of the vector induction functions obtained by reconstruction by applying the inverse Radon transform to the measurement nnym projections magnetic flux obtained using magneto-working body during its movement controlled spatial volume in the measurement method of forming a parallel projection data source.

The magnetically sensitive working element is a flat circuit based on an inductor. Moreover, its dimensions are set in such a way that, regardless of its position in the measurement volume, the initial vector field function decreases rather quickly at the boundaries of this contour.

, и проходящей на расстоянии s от начала системы координат объема измерения:The scalar value of the magnetic flux Φ _{D of the} initial vector function of magnetic induction B (x, y, z, t) through the surface D formed by the plane of the contour is an integral over the plane perpendicular to its normal vector $$\overline{n}$$

, and passing at a distance s from the origin of the coordinate system of the measurement volume:

$$F_D={\displaystyle \underset{D\left(s,\overline{n}\right)}{\int }\overline{B}d\overline{\sigma },\left(\mathrm{one}\right)}$$

- вектор нормали, выставленный к элементарной площадке плоскости интегрирования и численно равный ее площади. Переменные $$\overline{n}$$

и s задают положение контура в объеме измерения, причем в метрике сферической системы координат вектор нормали $$\overline{n}=\left(n,\theta ,\alpha \right)$$

.Where $$d\overline{\sigma }$$

is the normal vector set to the elementary area of the integration plane and numerically equal to its area. Variables $$\overline{n}$$

and s determine the position of the contour in the measurement volume; moreover, in the metric of the spherical coordinate system, the normal vector $$\overline{n}=\left(n,\theta ,\alpha \right)$$

.

Technically, magnetic flux is recorded due to the Faraday law, according to which the voltage u (t) induced in the circuit is determined by the expression:

$$u\left(t\right)=-\frac{d{F}_D\left(t\right)}{dt},\left(2\right)$$

where dФ _D (t) is the differential of the flux of the vector induction function; dt is the time differential.

Due to the introduced condition of periodicity in time, according to which the field accurately repeats its instantaneous states at equal time intervals equal to the period, the duration of the registration of one instantaneous volumetric state of the distribution of the magnetic field is no longer limited in time to the duration of the sampling interval, because the state can be recorded by measuring in space at time intervals that are multiples of the period. And despite the fact that the measurements are significantly spaced in time, all of them will be made for any one instantaneous state on the period.

Integration over time t of the differential dФ _D (t) expressed from equality (2) leads to the recording of the magnetic flux value:

$$F_D\left(t\right)=-{\displaystyle \underset{nT}{\overset{nT+t}{\int }}u\left(t\right)dt,\left(3\right)}$$

where T is the period during which one complete change in magnetic induction occurs; n = 1, 2, ... is the number of the period.

по оси OS и поворотов направления этих перемещений, задаваемых зенитным θ и азимутальным α углами сферической системы координат. Таким образом, дискретные параллельные перемещения многократно повторяются под разными углами, причем для зенитного в интервале от 0 до ½π, а для азимутального в интервале от 0 до π. В ходе управляемого пространственного перемещения рабочего органа после каждого его дискретного передвижения вдоль оси OS для текущего значения переменной положения s под углами, определенными вектором нормали $$\overline{n}$$

, регистрируется плоскостная проекция p - значение интеграла исходной функции индукции В по плоскости D, численно равное с учетом (1) значению пронизывающего магнитного потока Ф_D(t):The measurement begins with the procedure of controlled spatial movement of the working body in the measurement volume to obtain projections that correspond to the direct Radon transform for three-dimensional space (Fig. 1). Controlled spatial displacement implements a method of parallel formation of the initial projection data, for which the working body performs translational-rotational motion, which involves the alternation of discrete parallel displacements in the direction of the normal vector of the organ $$\overline{n}$$

along the OS axis and rotations of the directions of these movements, given by the zenith θ and azimuthal α angles of the spherical coordinate system. Thus, discrete parallel displacements are repeated many times at different angles, moreover, for the zenith in the interval from 0 to ½π, and for the azimuthal in the interval from 0 to π. During the controlled spatial movement of the working body after each discrete movement along the OS axis for the current value of the variable position s at angles determined by the normal vector $$\overline{n}$$

, a plane projection p is recorded - the value of the integral of the initial induction function B along the plane D, numerically equal, taking into account (1), the value of the penetrating magnetic flux Ф _D (t):

$$p\left(s,\overline{n},t\right)={\displaystyle \underset{D\left(s,\overline{n}\right)}{\int }\overline{B}d\overline{\sigma }=F_D\left(t\right).\left(\mathrm{four}\right)}$$

с учетом введенной зависимости от переменной времени записывается посредством интегралов по времени напряжений, индуцируемых изменением магнитного потока в соответствии с законом электромагнитной индукции Фарадея в контуре:Taking into account expression (3), based on (4), the projection of the values of the integral of the original induction function $$p\left(s,\overline{n},t\right)$$

taking into account the introduced dependence on the time variable, it is written by the time integrals of the voltages induced by the change in the magnetic flux in accordance with the Faraday law of electromagnetic induction in the circuit:

$$p\left(s,\overline{n},t\right)={\displaystyle \underset{D\left(s,\overline{n}\right)}{\int }Bd\overline{\sigma }=-{\displaystyle \underset{nT}{\overset{nT+t}{\int }}u\left(s,\overline{n},t\right)dt.\left(5\right)}}$$

Equation (5) is given for a spherical coordinate system, while in the Cartesian coordinate system (X, Y, Z) the expression of the components of this equality has the form:

$$\{\begin{array}{l}{p}_x\left(s,\overline{n},t\right)=\left[\sin \theta \cdot p\left(s,\overline{n},t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right)\right]\cos \alpha -\\ -\left[\sin \theta \cdot p\left(s,{\overline{n}}_{\perp \alpha },t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)\right]\sin \alpha ;\\ p_y\left(s,\overline{n},t\right)=\left[\sin \theta \cdot p\left(s,\overline{n},t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right)\right]\sin \alpha +\\ +\left[\sin \theta \cdot p\left(s,{\overline{n}}_{\perp \alpha },t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)\right]\cos \alpha ;\\ p_z\left(s,\overline{n},t\right)=\cos \theta \cdot p\left(s,\overline{n},t\right)-\sin \theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right),\end{array}\left(6\right)$$

, $${\overline{n}}_{\perp \alpha }=\left(n,\theta ,\alpha +\pi /2\right)$$

и $${\overline{n}}_{\perp \theta ,\alpha }=\left(n,\theta +\pi /2,\alpha +\pi /2\right)$$

- векторы нормалей плоскостей, соответственно перпендикулярно повернутых по углам θ и α на величину равную π/2 относительно текущего вектора нормали $$\overline{n}$$

.Where $${\overline{n}}_{\perp \theta }=\left(n,\theta +\pi /2,\alpha \right)$$

, $${\overline{n}}_{\perp \alpha }=\left(n,\theta ,\alpha +\pi /2\right)$$

and $${\overline{n}}_{\perp \theta ,\alpha }=\left(n,\theta +\pi /2,\alpha +\pi /2\right)$$

are the normal vectors of the planes, respectively, perpendicularly rotated in the angles θ and α by an amount equal to π / 2 relative to the current normal vector $$\overline{n}$$

.

Given the expression (5), the system of equations (6) is written in the form:

$$\{\begin{array}{l}{p}_x\left(s,\overline{n},t\right)=-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\cos \alpha +\\ +\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\sin \alpha ;\\ p_y\left(s,\overline{n},t\right)=-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\sin \alpha -\\ -\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\cos \alpha ;\\ p_z\left(s,\overline{n},t\right)=-\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt.\end{array}\left(7\right)$$

Cartesian components of the distribution of the vector induction function in the measurement volume are obtained by applying the reconstruction algorithm to the initial projection data (7) obtained during the controlled spatial displacement of the working body (Fig. 2).

The task of reconstructing the Cartesian components in the non-destructive volumetric measurement method is reduced to solving the basic integral equation (7) with finding the distribution of the components B _x , B _y , B _z according to the measured values of the planar projections p _x , p _y , p _z . The solution of the main integral equation (7) involves the use of a reconstruction algorithm based on the inverse Radon transform by means of back projection with Fourier filtering. Fourier filtering is carried out by convolving the projections of the components directly in the space of the original Fourier transform with the corresponding filtering space of the original Fourier transform with the corresponding filtering convolution function h (s), which is the inverse Fourier transform F ^-1 [] of the square of the frequency K of the spatial spectrum [6]. So from (7) get:

$$\{\begin{array}{l}{B}_x\left(x,y,z,t\right)={\displaystyle \int \left(p_x\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_x\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}[{\displaystyle \underset{0}{\overset{\pi }{\int }}((-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\cos \alpha +}}\\ +\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\sin \alpha )\ast h\left(s\right))d\alpha ]\cos \theta d\theta ;\\ B_y\left(x,y,z,t\right)={\displaystyle \int \left(p_y\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_y\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}[{\displaystyle \underset{0}{\overset{\pi }{\int }}((-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\sin \alpha +}}\\ +\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\cos \alpha )\ast h\left(s\right))d\alpha ]\cos \theta d\theta ;\\ B_z\left(x,y,z,t\right)={\displaystyle \int \left(p_z\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_z\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(-\left[\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt-\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta ,}\end{array}\left(8\right)$$

, $${\displaystyle \int d\overline{n}={\displaystyle \int {\displaystyle \int d\alpha cos\theta d\theta }}}$$

- дифференциал вектора нормали; dθ и dα - дифференциал зенитного и азимутального углов, соответственно; B_x(x,y,z,t), B_y(x,y,z,t), B_z(x,y,z,t) - x-,y-, z-компоненты векторной функции магнитной индукции $$\overline{B}$$

, соответственно.where the symbol "*" is a convolution operator; $$h(s)=F^{-\mathrm{one}}\lfloor K^2\rfloor ={\displaystyle \int K^2\cdot e^{2\pi Ks}dK}$$

, $${\displaystyle \int d\overline{n}={\displaystyle \int {\displaystyle \int d\alpha cos\theta d\theta }}}$$

- the differential of the normal vector; dθ and dα are the differential of the zenith and azimuthal angles, respectively; B _x (x, y, z, t), B _y (x, y, z, t), B _z (x, y, z, t) - x-, y-, z-components of the vector function of magnetic induction $$\overline{B}$$

, respectively.

Thus, the proposed method allows to obtain, in places inaccessible to mechanical penetration, in the scope of measuring the distribution of the components of the vector function of the magnetic induction nonuniformly distributed in space and periodically changing in time of the magnetic field, reconstructed by applying the principle of the inverse Radon transform to the measured magnetic flux projections obtained by controlled spatial movement of the working body, integrating the vector nktsiyu field.

Literature

1. USSR author's certificate No. 1652951, cl. G01R 33/02, publ. 05/30/1991.

2. USSR Copyright Certificate No. 1762282, cl. G01R 33/02, publ. 09/15/1992.

3. Copyright certificate of the USSR No. 1684761, cl. G01R 33/06, publ. 10/15/1991.

4. RF patent No. 2179323, cl. G01R 33/02, publ. 02/10/2002.

5. RF patent No. 2174235, cl. G01R 33/02, publ. 09/27/2001.

6. J. Radon. Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten // Berichte Sachsische Akademie der Wissenschaften, Bande 29, s.262-277, Leipzig, 1917.

Claims (1)

A method of non-destructive volumetric measurement of the vector function of magnetic induction non-uniformly distributed in space and periodically changing in time of a magnetic field, which consists in measuring the distribution of the vector function of magnetic induction of a periodically changing field at specific points in the space under study for arbitrarily selected time instants over a period and based on translational movements and rotations at the corners of the working magnetometer and the register induction of stresses induced in it, characterized in that the measurements of instantaneous volumetric states of the distribution of a nonuniform magnetic field in space are carried out in places inaccessible to mechanical penetration, and the working magneto-measuring organ is performed in the form of a space-orientated one contour connected to a spherical coordinate system, and during its controlled spatial displacement, a method for parallel formation of the initial projection data of the induction function, for which the working body makes a movement that involves alternating discrete parallel displacements in the direction of the axis of the normal vector of the organ and rotations of the directions of these displacements specified by the zenith and azimuthal angles of the spherical coordinate system, so that discrete parallel displacements are repeated many times at different angles, and for the zenith in the interval from 0 to $$\frac{\mathrm{one}}{2}\pi$$

, and for the azimuthal in the range from 0 to π, and the initial projection data of the Cartesian components of the distribution of the vector function of magnetic induction in the measurement volume in the Cartesian coordinate system necessary for the reconstruction algorithm are obtained by trigonometric transformations:
$$\{\begin{array}{l}{p}_x\left(s,\overline{n},t\right)=\left[\sin \theta \cdot p\left(s,\overline{n},t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right)\right]\cos \alpha -\\ -\left[\sin \theta \cdot p\left(s,{\overline{n}}_{\perp \alpha },t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)\right]\sin \alpha ;\\ p_y\left(s,\overline{n},t\right)=\left[\sin \theta \cdot p\left(s,\overline{n},t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right)\right]\sin \alpha +\\ +\left[\sin \theta \cdot p\left(s,{\overline{n}}_{\perp \alpha },t\right)+\cos \theta \cdot p\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)\right]\cos \alpha ;\\ p_z\left(s,\overline{n},t\right)=\cos \theta \cdot p\left(s,\overline{n},t\right)-\sin \theta \cdot p\left(s,{\overline{n}}_{\perp \theta },t\right),\end{array}$$

Where $$p\left(s,\overline{n},t\right)$$

- initial projection data of the induction function;
s is the coordinate of the direction axis of the controlled spatial movement of the working body;
θ and α are the zenith and azimuthal angles of deviation of the working body, respectively;
$${\overline{n}}_{\perp \theta }=\left(n,\theta +\pi /2,\alpha \right)$$

, $${\overline{n}}_{\perp \alpha }=\left(n,\theta ,\alpha +\pi /2\right)$$

and $${\overline{n}}_{\perp \theta ,\alpha }=\left(n,\theta +\pi /2,\alpha +\pi /2\right)$$

are the normal vectors of the planes, respectively, perpendicularly rotated at angles θ and α by an amount equal to π / 2 relative to the current normal vector of the working body $$\overline{n}$$

;
t is the time;
p _x , p _y , p _z are the initial projection data of the x-, y-, z-components of the vector function of magnetic induction, which are written through magnetic fluxes, numerically equal to the time integrals of the stresses induced in the circuit:
$$\{\begin{array}{l}{p}_x\left(s,\overline{n},t\right)=-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\cos \alpha +\\ +\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\sin \alpha ;\\ p_y\left(s,\overline{n},t\right)=-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\sin \alpha -\\ -\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\cos \alpha ;\\ p_z\left(s,\overline{n},t\right)=-\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt,\end{array}$$

where u (t) are the stresses induced by a change in the flow in the circuit at time t;
T is the period during which one complete change in the function of magnetic induction occurs;
n = 1, 2, ... is the period number, and the inverse Radon transform based on their convolution filtering using the convolution function, which is the inverse Fourier transform of the square of the frequency of the spatial spectrum, is applied to the initial projection data
$$\{\begin{array}{l}{B}_x\left(x,y,z,t\right)={\displaystyle \int \left(p_x\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_x\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}[{\displaystyle \underset{0}{\overset{\pi }{\int }}((-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\cos \alpha +}}\\ +\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\sin \alpha )\ast h\left(s\right))d\alpha ]\cos \theta d\theta ;\\ B_y\left(x,y,z,t\right)={\displaystyle \int \left(p_y\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_y\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}[{\displaystyle \underset{0}{\overset{\pi }{\int }}((-\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\sin \alpha +}}\\ +\left[\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \alpha },t\right)dt+\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta ,\alpha },t\right)dt\right]\cos \alpha )\ast h\left(s\right))d\alpha ]\cos \theta d\theta ;\\ B_z\left(x,y,z,t\right)={\displaystyle \int \left(p_z\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\overline{n}={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(p_z\left(s,\overline{n},t\right)\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta =}}\\ ={\displaystyle \underset{0}{\overset{\pi /2}{\int }}\left[{\displaystyle \underset{0}{\overset{\pi }{\int }}\left(-\left[\cos \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,\overline{n},t\right)dt-\sin \theta \cdot {\displaystyle \underset{nT}{\overset{nT+t}{\int }}u}\left(s,{\overline{n}}_{\perp \theta },t\right)dt\right]\ast h\left(s\right)\right)d\alpha }\right]\cos \theta d\theta ,}\end{array}$$

where the symbol "*" is a convolution operator;
h (s) is a convolution function;
$$d\overline{n}$$

- the differential of the normal vector;
dθ and dα are the differentials of the zenith and azimuthal angles, respectively;
B _x (x, y, z, t ₀ ), B _y (x, y, z, t ₀ ), B _z (x, y, z, t ₀ ), where x, y, z are the components of the magnetic vector function induction B, respectively, due to which the Cartesian components of the distribution of the vector function of magnetic induction in space are reconstructed.

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