Abstract
A mathematical model is developed to study the operation of three-phase asynchronous motor with squirrel-cage rotor when the stator winding is powered from a single phase network. To create a rotating magnetic field one of the phases is fed through the capacitor. Due to the asymmetry of power feed not only transients, but the steady-state regimes are dynamic, so they are described by differential equations in any coordinate system. Their study cannot be carried out with sufficient adequacy on the basis of known equivalent circuits and require the use of dynamic parameters. In the mathematical model the state equations of the circuits of the stator and rotor are composed in the stationary three phase coordinate system. Calculation of the established mode is performed by solving the boundary problem that makes it possible to obtain the coordinate dependences over the period, without calculation of the transient process. In order to perform it, the original nonlinear differential equations are algebraized by approximating the variables with the use of cubic splines. The resulting nonlinear system of algebraic equations is a discrete analogue of the initial system of differential equations. It is solved by parameter continuation method. To calculate the static characteristics as a function of a certain variable, the system is analytically differentiated, and then numerically integrated over this variable. In the process of integration, Newton's refinement is performed at each step or at every few steps, making it possible to implement the integration in just a few steps using Euler's method. Jacobi matrices in both cases are the same. To account for the current displacement in the rods of the squirrel-cage rotor, each of them, along with the squirrel-cage rings, is divided in height into several elements. This results in several squirrel-cage rotor windings which are represented by three-phase windings with magnetic coupling between them.
Highlights
Трехфазные асинхронные двигатели (АД) в случае обрыва одной из фаз сети могут работать как однофазные с меньшей нагрузкой, однако после остановки не могут быть пущены в ход без принятия дополнительных мер, поскольку обмотки двух оставшихся в работе фаз при однофазном питании создают не вращающееся, а пульсирующее магнитное поле [1, 2]
A mathematical model is developed to study the operation of three-phase asynchronous motor with squirrel-cage rotor when the stator winding is powered from a single phase network
a rotating magnetic field one of the phases is fed through the capacitor
Summary
Наиболее полно электромагнитные процессы в АД можно исследовать в трехфазной системе координат, однако в естественных физических осях дифференциальные уравнения (ДУ) содержат периодические взаимные индуктивности. От этих проблем можно избавиться путем перехода к неподвижной трехфазной системе координат [16], в которой обмотка статора остается непреобразованной, а вращающийся ротор заменяется заторможенным. С целью учета вытеснения тока в стержнях и насыщения магнитопровода ротора потоками рассеивания пазовая часть стержней, а также короткозамыкающие кольца разбиваются по высоте на k слоев В результате получим на роторе k короткозамкнутые обмотки, которые преобразуем к трехфазным в соответствии с общепринятой методикой [2]. При этом в математической модели АД рассматриваются следующие неподвижные трехфазные обмотки 3): статора – A, B, C и k обмоток – al ,bl ,cl (l = 1, ..., k) ротора [13,14,15,16]
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