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The Numerical Modelling and residual field computation of flat ferromagnet at its magnetization by inhomogeneous pulse magnetic fieldMatyuk V.F., Churilo V.R., Strelyukhin A.V.
Institute of Applied Physics of National Academy of Sciences of Belarus
Minsk, Republic Belarus
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
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
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
The boundary conditions
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.
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  were used.
The components c in the elements of division by the Biot-Savare-Laplace Law  determine.
For computation the value of magnetization in each element of division in every time interval the nonlinear integral equation  was solved
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
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
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.
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.
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