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The purpose of this article is to develop a method and algorithm for the calculation of the steady-state mode and static characteristics of the squirrel-cage asynchronous motors that are supplied by a three-phase network with an asymmetrical voltage system. The aim is achieved by developing a method where the mathematical model of the motor takes into account the saturation of the magnetic core and the current displacement in the short-circuited rotor bars. Magnetization characteristics of the main magnetic flux and the leakage fluxes paths are used to account for the saturation. To account for the current displacement in the rotor winding the bars are divided into several layers throughout the height, resulting in a number of windings on the rotor with mutual inductive couplings. In the mathematical model, the processes in the asynchronous motor are described by a system of the nonlinear differential equations of electromagnetic equilibrium, composed in the fixed three-phase coordinate axes. The solution of this system in the steady-state mode is the periodic dependence of the currents, which are determined by solving the boundary problem. For this purpose, their algebraization is carried out by approximation of the periodic dependences at the nodes’ grid of the period of repeatability of the process by the third order splines. The solution of the problem is to determine the nodal values of coordinates on the period from the obtained nonlinear system of algebraic equations. For this purpose, the continuation method and iterative refinement by the Newton method were used. Calculation of the static characteristics was performed by the differential method.

Цель. Целью статьи является разработка методики применения аналитической записи кусочно-линейной аппроксимации кривой намагничивания для расчета нелинейной электрической цепи, содержащей трансформатор, работающий в режиме насыщения магнитопровода. Методика. При разработке методики кусочно-линейная аппроксимация и нелинейное дифференциальное уравнение электрической цепи записывались с использованием единичных разрывных функций. Расчет нелинейной задачи для нахождения тока в трансформаторе был аналогичен нелинейной задаче для определения напряженности магнитного поля в ферромагнетике с кусочно-линейной аппроксимацией кривой намагничивания. Результаты. Разработана методика расчета токов и напряжений трансформатора, работающего в режиме насыщения, на основе аналитической записи кусочно-линейной аппроксимации кривой намагничивания магнитопровода. Научная новизна. Впервые применительно к характеристике намагничивания пик-трансформатора использована аналитическая запись кусочно-линейной аппроксимации. Практическое значение. Предлагаемая методика может быть непосредственно использована для электромагнитного расчета вариантов пик-трансформатора в составе общего (управляемого) электротехнического устройства на стадии проектирования и оптимизации. The purpose. Paper purpose is working out of a technique of the solution of nonlinear problems in an analytic form, namely a piecewise linear ap-proximation of magnetization curve of a magnetic circuit by discontinuous functions, and the solution of the nonlinear differential equation for an electric circuit which contains this type of device. For an example of use of a technique the approximate calculation of the peak-transformer working in a regime of saturation of a magnetic circuit is resulted. Methodology. By working out of a technique of piecewise linear approximation in the capacity of the basic mathematical apparatus individual discontinuous functions were used. Calculation of a nonlinear problem for a current finding in the transformer was analogous to a nonlinear problem for magneticfield strength finding in a ferromagnetic. Results. The design procedure of cur-rents and voltage of the transformer working in a regime of saturation, on the basis of analytical record of a piecewise linearapproximation of magnetization curve of a magnetic circuit is developed. Originality. For the first time with reference to the characteristic of magnetization of the peak-transformer the method of piecewise linear approximation is applied. Practical value. The offered technique can be directly used for electromagnetic calculation of peak - transformer as a part of the general (operated) electrotechnical device at a design stage and optimization.

Описана методика синтеза математической модели асинхронного двигателя. Предлагается выполнять эксперименты в следующей последовательности. В среде AutoCAD строятся геометрические модели, после импорта которых в Comsol Multiphysics и последующей доработки под управлением Matlab производится расчет коэффициентов и зависимостей, которые используются в математической модели асинхронного двигателя. A methodology of an asynchronous motor mathematical model synthesis is described. Experiments are suggested to be conducted in the following sequence. Geometrical models are first built in AutoCAD, then imported to Comsol Multiphysics, and further processed in Matlab with computation of coefficients and dependences applied in the asynchronous motor mathematical model.