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## The Numerical Modelling and residual field computation of flat ferromagnet at its magnetization by inhomogeneous pulse magnetic field

Matyuk V.F., Churilo V.R., Strelyukhin A.V.
Institute of Applied Physics of National Academy of Sciences of Belarus
Minsk, Republic Belarus
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### Introduction

The pulse magnetic method [1] of nondestructive testing of ferromagnetic articles is widely used in steel plants of CIS, Germany, Japan and other countries. As the informative parameter the gradient of the residual field magnetization at local magnetization is used. The further development of that method requires the detailed study of processes of local magnetization of ferromagnetic articles in the nonuniform pulse magnetic field. The complexity of analysis of that problem [2] and the present-day state of computers design take up to use of numerical methods. In this paper the two-dimensional mathematical model and the procedure of numerical solution of the residual field magnetization of flat ferromagnet in the form of disk magnetized in the pulse magnetic field of superimposed transducer is discussed.

### Model description

As the object of investigation the flat ferromagnet in the form of disk magnetized in the field of small-sized solenoid putted on one from its surfaces is taken.

By model construction the next assumptions were made: the test piece initially demagnetized is, the material isotropic is, the magnetic ductility absent is. The dependence of magnetic properties on the external field by the known approximations for static magnetic characteristics were determined. By solution the discrete model as in the space (the model of ferromagnet with piece-permanent approximation of magnetization on the elements of division), as in time is used.

The general rules of distribution of magnetic field in the article by the system of Maxwell equations can be described the solving of what in our case to the nonlinear vector equation of parabolic type can be reduced

 (1)

where the magnetic field strength is; md(H) the incremental permeability is;l the electrical conductivity of the material of disk is.

Because the cylindrical symmetry of the problem its further solution in the cylindrical coordinates system (Hj = 0) was made. Then for components (Hr,Hz) the equation (1) becomes

 (2)

As in the process of pulse magnetization the field on the surface of ferromagnet not only by the field of solenoid, but by the field of magnetization of the article itself and by the field of eddy currents determined is it was suggested to use the next initial and boundary conditions allowing to take into account the contribution in the field on the boundary of the three above discussed components.

The initial condition

 (3)

The boundary conditions

 (4)

where G* is the portion of the boundary G of the rectangle that haven't the side r = 0; H*r, H*z is the field on the boundary G* including the field of solenoid c, field of ferromagnet M and field of eddy currents d .

Consider that the field of solenoid c at every instant of time tl known is, and the components M and d from the solution (2) on the preceding time interval tl-1 determined are.
Then

 (5)

For determination the magnetic state of separate elements of division in dependence on the increase or the decrease of the field strength H in that element the equations for m d(H) from paper [3] were used.

The components c in the elements of division by the Biot-Savare-Laplace Law [4] determine.

For computation the value of magnetization in each element of division in every time interval the nonlinear integral equation [4] was solved

 (6)

where κ the magnetic susceptibility is; VM the volume of the magnetized material (element) is; Q the observation point is; N the source point (the point determining the coordinates of element of division of ferromagnet) is; the radius-vector from the source point (N) into the observation point (Q) is.

In connection with the nonlinear dependence of magnetic susceptibility on the magnetic field strength the solution (6) is realized as reduction to the system of linear equations and using the method of iteration.

The computation of contribution of magnetic field strength on the boundary due to the eddy currents on the base of finite differential approximation of equation is performed

 (7)

and the components of magnetic field using the Biot-Savare-Laplace Law were determined.

On the computed in the end of pulse remanent magnetization values using equations

 (8)

where Z1, Z2, R1, R2 the geometrical coefficients are; n = 1,2,...N1 the number of the elements volume is,
the components of the residual field strength in the observation regions were determined.

### Verification of model

For checking the suggested model the computation and experimental investigation of residual field distribution of flat ferromagnet was curried out. For solving the system (2) the method of finite differences was used. At modeling the magnetizing pulse was assumed that the pulse rises on the sinusoidal law and decreases on the law exp(-t), where b the constant is. It corresponds to the form of magnetizing pulse by experimental checking. The accuracy of iteration for computation the magnetization 1% was given. The results of the investigations for disk with diameter of 195 mm from the steel with coercive force 760 A/m at different amplitudes and duration of pulse edges in the Figure are done. The magnetizing solenoid have dimensions: the inside radius is 5 mm, the outside radius is 25 mm, the length is 30 mm, the number of turns is 265.

 Fig 1: The distribution of normal Hrz(O) and tangential Hr r() components of residual field at pulse magnetization along the disk surface (the number of division was 5´50) in the obser-vation region from the side of magnetization (computation --, experiment -O-, --): Hm = 5·105 A/m; a) - t0 = 0,85 ms; t3 = 1 ms;        b) - t0 = 0,85 ms; t3 = 10 ms; c) - t0 = 0,2 ms; t3 = 6 ms;        d) - t0 = 1,2 ms; t0 == 6 ms; Hm is the pulse amplitude; t0 , t3 are the duration of the leading and the trailing edge of magnetizing pulse

The received results show that the considered model the real distribution of residual field magnetization after applying the nonuniform pulse magnetic field to the flat ferromagnet in the form of a disk represents. For more precise quantitative correspondence the further development of the method of computation, the increase of the number of division of space and time grid, specification of more precise computation of iteration procedures and use of more powerfull PC are required.

Thus the general rules of residual magnetization field distribution computed using the suggested method in type and in value coincide with the experimental data with acceptable for practice accuracy. This confirms the legitimacy of use the developed model for computation the magnetic state of ferromagnetic disk in the nonuniform pulse field of the superimposed solenoid.

### References

1. Melgui M.A. Magnetic testing of mechanical properties of steels. Minsk: Nauka i technika, 1980. - 184 p.
2. Zatsepin N.N. Nonlinear equations of magnetodynamics and magnetostatics of isotropic ferromagnetic medium placed in the nonuniform magnetic field. - Vesti AN BSSR. Ser. phys.-techn. nauk, 1984, No. 2, p.80-91.
3. Melgui M.A. Formulae for description nonlinear and hysteresis properties of ferromagnets. - Defectoskopy, 1987, No.11, p.3-10.
4. Chari M.V.K., Silvester P.P. Finite elements in electrical and magnetic field problems. - New York, Chichester, Brisbone, Toronto: John Willey&Sons, 1980. - 220 p

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