The object of research is the processes in magnetic systems with nonlinear ferromagnetic elements. The purpose of the work is improvement of the method of integral equations for the calculation of heterogeneous magnetic fields in nonlinear media in the direction of reducing the number of components in the cores of integral equations that contain the function of the gradient from the magnetic permeability, and the subsequent explicit expression of this function through the density of the sources of the magnetic field, and the application of these approaches for the calculation of magnetic systems specific electrical devices. Research methods: the solution of the tasks set in the work was carried out by methods based on the use of the theory of the electromagnetic field, vector analysis, the concept of secondary sources, the theory of integral equations, the theory of approximation, the theory of numerical integration, numerical solution of systems of integral equations, numerical solution of systems linear algebraic equations. Theoretical and practical results and novelty: the method of integral equations for calculating the characteristics of the magnetic field in non-linear media was further developed in the direction of reducing the number of components in the cores of the integral equations, which contain the function of the gradient from the magnetic permeability, and the further explicit expression of this function through the densities of the sources of the magnetic field , which makes it possible to replace the procedure of numerical differentiation when approximating the gradient function from the magnetic permeability with the procedure of numerical integration, which is of significant importance for increasing the accuracy of the calculation of the kernels of integral equations in the method of integral equations. A new mathematical model and corresponding calculation method for the numerical calculation of the magnetic field in a collectorless DC motor with an implicit-pole smooth stator containing thin ferromagnetic transitions between the stator winding and the gap between the stator and an explicit-pole rotor with permanent magnets, which allow taking into account the nonlinear properties of the ferromagnetic elements of the magnetic field, have been developed engine systems. Subject and degree of implementation: the proposed algorithm for calculating the magnetic field in a collectorless DC motor with an implicit-pole smooth stator, which contains thin ferromagnetic transitions between the stator winding and the gap between the stator and an explicit-pole rotor with permanent magnets, is of practical importance for the electrical engineering industry, which allows taking into account nonlinear properties ferromagnetic elements of the magnetic system of the engine. Based on it, a program was written in the FORTRAN language, which allows you to calculate the magnetic field in the working gap of the electric motor based on the known input data (geometric parameters of the system, electrophysical parameters of materials, characteristics of permanent magnets, current in the windings) and carry out optimization according to the selected criteria. Effectiveness of implementation: a technique for calculating the gradient function from the magnetic permeability was developed, taking into account the explicit representation of the magnetic field strength through the density of magnetic charges, which allowed the process of its approximation by a finite-dimensional analogue to be performed by integration over the field sources, which allowed to reduce the error of its approximation, in contrast to the procedure of numerical differentiation of the function directly magnetic permeability. Field of use: for the development of methods of calculating electromagnetic fields and theoretical electrical engineering is important for the field.
