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TWO SIDED METHOD FOR CALCULATION
OF DYNAMIC ELECTROTECHNICAL SYSTEMS
WITH DISTRIBUTED PARAMETERS WITH GIVEN THE ERROR OF THE ORIGINAL DATA
Tigran R. Harutyunyan, Moscow, Russia,
tigran 201094@ mail.ru
Abstract
Bilateral methods for calculating the magnetic field characteristics of electrical systems containing ferromagnets and permanent magnets are considered. The methods are based on the application of the Pontryagin maximum principle to the electromagnetic field equations in terms of scalar and vector potentials. The solution procedure uses the transition from the differential formulation of boundary value problems of the magnetic field to the corresponding discrete-continuous in the form of a system of ordinary differential equations, to which the classical theory of the maximum principle applies. After finding the equations of the boundary value problem of the maximum principle, the inverse limit transition to the differential form is carried out by means of the grid step striving to zero. The corresponding conjugate partial differential equations for different optimality criteria are obtained. The solution of the problem of calculation of bilateral estimates of the solution in the calculation of the magnetic field in a ferromagnetic placed in an external equal magnetic field is considered. This method is also applicable for the calculation of the fields of permanent magnets, which requires taking into account the residual magnetization, the final width of the hysteresis loop. For this purpose, the corresponding ratios are given. It is assumed that the main source of error in the calculation is the approximate values of the magnetic permeability of the medium.
In comparison with the used method of a small parameter, the limitations on the error of the parameters and characteristics of the boundary value problem are much smaller, which greatly expands the range of problems to be solved. Although an example of a plane-parallel field was considered in the article, the idea and the main relations of the method remain unchanged for the three-dimensional problems of the magnetic field theory. The proposed method has advantages in solving those problems in which it is required to find estimates of the solution in the neighborhood of a finite number of singular points. As shown, there is a fundamental possibility to estimate the mean square error of the solution in the entire region. The corresponding examples are the problems associated with switching overvoltages, current surges when switching circuits with nonlinear inductances, direct and inverse problems of calculating permanent magnets, when assessing the effect of the hysteresis loop width. The possibilities of modern numerical technique allow to find with the help of the considered methods also effective uniform estimates of the solution at large intervals. The method can be used to improve the reliability of the results in the design calculations of various electrical devices, as well as in the problems of the theory of magnetic measurements.
The obtained results can also be used in solving direct and inverse problems for the system of ferromagnetic bodies and in test problems using other methods. The developed approach can also be successfully applied in solving a wide range of dynamic problems for systems with distributed parameters, including the theory of elasticity, thermal conductivity, piezoelectric oscillations, quantum mechanics and other fields.
Keywords: calculation of magnetic field, two-way method, error of coefficients, ferromagnets, permanent magnets, scalar and vector magnetic potentials, maximum principle.
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