Modeling an induction motor based on the equations of a generalized electric machine, taking into account the saturation of the magnetic circuit

. Writing the differential equations of an asynchronous electric motor on the basis of a generalized electric machine in different coordinate axes can not only reduce the number of model equations, but also introduce various factors in the mathematical description that increase the modeling accuracy. It is proposed to write the equations of the mathematical model of an induction motor taking into account the saturation of the magnetic circuit and solve them using the Matlab application package. The proposed approach improves the accuracy of calculating the mechanical characteristics of an asynchronous electric drive.


Introduction
An important step in the synthesis and analysis of electric drive systems is the development of a model of an electric machine that meets the challenges of vector control [1][2][3][4][5][6].
In most cases, electromechanical energy converters are made in multiphase design. This fact complicates the mathematical apparatus that describes the transient processes in such converters. This is due to an increase in the number of equations of the mathematical model of the experimented electric machine. Therefore, as a rule, they strive to make an equivalent transition from a real machine to a generalized double-phased [7,8]. The generalized machine in the mathematical description is characterized by the following generally accepted assumptions: "the phase windings are symmetrical, the air gap is uniform, there are no losses in steel and tooth phenomena, the magnetizing circuit is linear, the magnetizing force of each winding and the magnetic induction in the air gap are sinusoidal around the circumference of the stator (rotor), rotor parameters are reduced to the stator circuit " [7][8][9]. The equations of the mathematical model of a generalized machine can be written in different forms depending on the coordinate systems used. The most commonly used coordinate systems are: u,v; α,β; d,q [7,8].

Methods
We write the equations of the generalized machine in the coordinate system u,v rotating synchronously with the speed of the electromagnetic field: -active resistance of stator and rotor windings, respectively; p -differentiation operator; n P -number of pole pairs. Equations of a generalized electric machine in a coordinate system u,v -most common. Transition from coordinates v,u to a fixed coordinate system α,β occurs by direct coordinate transformation at 0 k ω = .
Transition from coordinate system u,v to the coordinate system d,q rotating at the speed of the rotor, if we assume that R ω k ω = . From here, the system of phase voltages applied to the windings will be equal: .
The system of equations (1) corresponds to the structure of the electromechanical energy converter (EMF), shown in figure 1. This structure was compiled in the Simulink application of the Matlab application package [10][11][12]. Here, the voltages υ 2 are transformed control actions connecting the engine with the control system in the axes x, y. Transients in electromechanical converters are classified as follows: during start-up, braking, reverse, restart, and also when the load changes [13][14][15][16]. To obtain information about the features of transient processes in an electric machine, reliable data on the parameters and independent variables of the model are needed.
In most cases, the nature of transient processes in EMF affects the configuration of the drive as a whole. Namely: overall dimensions, permissible electromagnetic loads, operating modes, including the features of starting and braking these systems.    In order to optimize the EMF by mass and size indicators, the "operating point" is selected on a nonlinear section of the magnetization characteristic. When changing the parameters of the supply network, the degree of saturation of the magnetic circuit of the machine changes. This leads to the fact that the intrinsic and mutual inductances between the windings change, therefore, it is necessary to take into account changes in the inductive parameters of the electric machine over time [17,18]. Variation of these parameters, in turn, will lead to a variety of EMF characteristics, both in steady state and in transient modes.

Results
When starting the above-considered asynchronous electric motor due to the presence of alternating components in the amplitudes of the stator and rotor currents, and, accordingly, the magnetization current, the magnetic circuit of an electric machine can enter a deep saturation state. The inductance of the magnetization and scattering circuit of the windings of a saturated machine significantly differ from the inductances of the nominal mode, and with increasing saturation the inductance decreases. Within the limits of calculation error, it can be assumed that the magnetic flux of scattering closes through the air, therefore, we can assume that the inductive parameters of the stator and rotor will not depend on saturation.
The graphs of changes in inductance versus magnetization current will be obtained from the relationships shown in figure 4.
The magnetization current can be determined by adding the projections of the stator and rotor currents on the axis u,v. As  I  I  I  +  = , then from the vector diagram shown in figure 6, it can be seen that: The magnitude of the magnetization current from equation (5), (6) is determined by the following formula: The equations of an asynchronous machine, taking into account the saturation of the magnetic circuit, are written as follows   Graphs of transients of the magnetization current (curve 1) and magnetization inductance (curve 2) when starting an induction motor with a load on the shaft equal to the nominal value is shown in figure 9. Graphs of transients of speed (curve 1) and electromagnetic moment (curve 2), under similar conditions conducting experiments are shown in figure 10.   Fig. 9. Graphs of transients of the current and magnetization inductance at start, loaded with a rated torque, of the motor.

The discussion of the results
The availability of reliable information about the nature of the transition processes in modern electric drives makes it possible to increase the reliability and accuracy of the operation of critical actuators. For a more accurate mathematical description of transients, various generally accepted assumptions are removed. When trying to take into account an assumption, a differentiated approach is needed. The essence of this approach is that we take into account only those factors that, for the considered group of machines and their operating conditions, have a significant impact on the accuracy of the calculation of the mathematical model. In the framework of this work, the possibility of taking into account such an important factor as the saturation of the magnetic circuit in the calculation of transients in a specific induction motor using modern Matlab application software is shown.