Transformations of equations with approximation of characteristics of nonlinear elements by power polynomial

. The article discusses some transformations of the equation associated with approximations in the form of a polynomial, which is quickly transferred to a convenient equation compiled with respect to the main input and output values and gives generalized dependencies for a number of nonlinear systems. As an example, a part of a magnetic circuit with longitudinal and orthogonal magnetization fields is given. Relations are presented that connect the inductions B 1 , B 2 as input and output quantities, which in turn are directly related to the corresponding voltages U 1 and U 2 . For all three magnetic circuits, dependencies are obtained that convert the input and output currents i 1 , i 2 into input and output inductions B 1 , B 2 with an approximating function in the form of a polynomial of any degree.


Introduction
When studying processes in electrical circuits with nonlinear elements, in some cases the problem arises of transforming equations that include approximations of the characteristics of non-linear elements.These transformations are often reduced to obtaining such dependencies (usually connecting the main initial and input values) that are convenient in the study of amplitude, frequency, phase, and other characteristics.However, such transformations often cause great difficulties inherent in the operation of non-linear dependencies [1][2][3][4].
The article discusses some transformations of the equation associated with an approximation in the form of a polynomial, which is sufficient to quickly transfer to a convenient equation compiled with respect to the main input and output quantities and give generalized dependencies for a number of nonlinear systems [4][5].

The current state of the investigated problem
As an example, three magnetic circuits are given (Fig. 1, 2, 3), which are parts of certain electrical circuits.The material of ferromagnetic cores is assumed to be isotropic and the same for all three magnetic circuits, and the dependence of the induction on the magnetic field strength (magnetization curve) is unambiguous and symmetrical relative to the origin of coordinates [6][7][8].As is known, such a magnetization curve can be approximated by a polynomial: (1) Where: Н -magnetic field strength; m -counted integer; В -magnetic field induction; a 1 , a 3 , a 5 -approximation coefficients., 2 show magnetic circuits with longitudinal magnetic circuits of magnetization.The end rods of a three-rod chain are assumed to be the same.The same condition (identity) is imposed on the cores of a twocore circuit.Figure 3 shows a magnetic circuit with orthogonal magnetic fields of magnetization [9][10][11][12].
Equation transformations are made for the general case when any energy sources function in primary and secondary electrical circuits (in one of the electrical circuits, instead of sources, autoparametric excitation of oscillations at any frequency can take place) [13][14].
Let us consider the functional connections of a threerod magnetic circuit (Fig. 1): (2) (3) Where: -strength of the magnetic field of the extreme rods; -the lengths of the extreme and rods, respectively.

Conclusion
In conclusion, the following conclusions can be drawn: 1.For all three magnetic circuits, dependences are obtained that convert the input and output currents i 1 , i 2 into input and output inductions B 1 , B 2 about an approximating function in the form of a polynomial of any degree; 2. Relations ( 13), ( 14) of a three-rod magnetic circuit, (17), ( 18) of a two-core one, ( 21), ( 22) -circuits with orthogonal fields differ only in the form of the coefficients (the term in ( 14) is combined with the first term at K=0, which makes it possible to obtain generalized dependencies for similar magnetic circuits that are parts of certain electrical circuits