Induction Motor Computer Models in Three-Phase Stator Reference Frames: A Technical Handbook

By Mikhail Pustovetov, Konstantin Shukhmin and Sergey Goolak

This book is a compilation of knowledge about computer models in the three-phase stator reference frame. Chapters explore several aspects of the topic and build upon research previously presented by contributors. The book aims to provide interesting solutions to problems encountered in the design of railway and analysis in railway motors. The modeling approaches proposed by the authors in this book may become an incentive for readers and researchers to develop their ‘lifehacks’ to solve new problems in induction motor design and testing.

Key topics presented in the book:
- Approximate calculations of induction motor equivalent T-shaped circuit parameters with the use of catalogue data
- Simulations of different types of shaft load, including fluid coupling
- Receiving static characteristics of an electric machine during simulation by means of dynamic model
- Simulation of the electric drive’s specific applications with three-phase induction motors building
- Direct start of an induction motor as part of an auxiliary drive of an AC electric locomotive, containing a capacitor phase splitter, starting with different types of shaft loads (fan or compressor).

Special attention has been given to the description of the thermal model of an induction motor with a squirrel-cage rotor, which makes it possible to simulate operating modes when powered by an unbalanced voltage, as well as with squirrel cage defects. The thermal model is presented as a detailed superstructure to the model of electromechanical processes of an induction electric machine.

Other key features of the book include references for further reading, an appendix for the parameters of the equivalent thermal circuit of an NVA-55 induction motor.

The material presented in the book is of interest to railway motor engineers, specialists in electromechanics and electric drives who use SPICE-compatible CAD applications in their work.

• Language: English
• Publisher: Bentham Science Publishers
• Release date: Oct 29, 2009
• ISBN: 9789815124309

Book preview

A Use of Nonlinear Coefficients in Ordinary Differential Equations of Mathematical Models of Electrical Devices Describing Inductance vs Current or Magnetic Flux Linkage Relationships

Mikhail Pustovetov¹, *, Konstantin Shukhmin²

¹ RIF Shipyard, Rostov-on-Don, Russia

² EIM Training Pty Ltd, Cairns, Australia

Abstract

When mathematical models of electromagnetic devices are described by ordinary differential equations, then the magnetization curves of their cores are often expressed with a help of nonlinear coefficients. Sometimes such an approach is considered questionable, and therefore, the purpose of this chapter is to prove its admissibility.

Keywords: Choke, Induction motor, Magnetization curve, Mathematical model, Nonlinearity, Ordinary differential equations, Transformer.

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* Corresponding author Mikhail Pustovetov: RIF Shipyard, Rostov-on-Don, Russia; Tel: [phone], E-mail: [e-mail]

INTRODUCTION

Some or even all parameters in the equations of an induction motor (IM)/drive mathematical model (MM) can vary over time or other characteristics of the system. In the equations of electromechanical energy conversion, it is common to describe coefficients in front of the independent variables as parameters [1]. Herein the term parameters is used for the inductances and resistances of the equivalent circuit and combined moment of inertia of the rotating parts on the motor’s shaft. Depending on the degree of precision or abstraction of the MM, the values of some and even all coefficients could be set as constants. Some of the IM practical problems were described and solved in [2] and [3] with a help of MM where all of the parameters were constants or allowed for the curvature of magnetic saturation along the main magnetic flux path.

Often used in ordinary differential equations (ODE), nonlinear coefficients describe inductance in electromagnetic devices (electrical machines [4, 5], transformers [6, 7], and chokes [3] vs current or magnetic flux linkage.

Sometimes the validity of such approximation of the magnetization curves is questioned by researchers dealing with MM of electromagnetic devices. A simple answer to these doubts is just an absence of data required for precise equations of the magnetic curves.

COMMON CASE

A time derivative of the magnetic flux linkage is:

Quite often the equation is solely influenced by the magnetic saturation of cores upon their permeability.

And for this particular reason in MMs:

Magnetically symmetric structures are used for 3-phase IM and transformers;

In IMs, smooth airgap surfaces are considered, neglecting any effects of magnetic field serration;

Adopted coordinate systems in IM are stationary which ensures the absence of the periodic coefficients [1, 4, 8, 9]. Those coefficients would allow for periodic changes in the mutual inductance of the phases during the rotation of the rotor.

Choke’s core and other parts are assumed as stationary.

The inductance below depends on the absolute value of current or magnetic flux linkage and can be expressed as the sum of constant and variable components:

therefore

Due to the inverted symmetry of the magnetization curve we can conclude that in a steady-state period of alternating current, its instantaneous value has double the number of the sine wave periods in the variable component of the inductance corresponding to the magnetization symmetry.

Oscillations of the magnetization current of IMs and transformers due to the filtering properties of inductors in general and a large value of the main inductance are basically sinusoidal even when supplied with a rectangular form of PWM voltage.

Fig. (1.1) [3] shows an example of transient changes in current and inductance of a choke with a magnetic core after the moment when its AC voltage supply is turned ON.

Assuming that the oscillations of the current have similar frequency and shape as the variable component of the inductance, then

Then

Fig. (1.1))

Simulation results for the RS38 choke when a sinusoidal voltage is applied to the terminals [3].

Thus it is fair to interpret the Lvar multiplier in the second constituent of the (1.5) as a nonlinear coefficient which is derivative of current or magnetic flux linkage.

Such interpretation of Lvar will lead to the ratio (1.6) which is used in a number of studies related to the MM of IM, transformers and chokes:

SPECIAL CASE

The following ratio (1.7) is specially derived for MMs of IMs:

where the inductance is a function of the absolute instantaneous value of the amplitude depicting the vector of magnetic linkage and in literature [2], it is defined in Cartesian plane as:

It should be noted that the magnetic flux linkage expressed by (1.8) under the symmetry conditions of the IM and its power supply changes significantly slower than the supply voltages and currents.

This is demonstrated by Fig. (1.2) showing transient start-up simulated process of a three-phase IM (Type AZHV250M2RUHL2) supplied by a three-phase, balanced and pure sin-wave voltage system.

This IM is used as a fan-motor on board of the Russian passenger mainline electric locomotives EP200.

The following designations are used in Fig. (1.2) and Fig. (1.3):

Lμ - the main inductance of IM;

Ψμ - the instantaneous amplitude of the magnetic flux linkage depicting vector of the IM mutual induction.

In case of IM, we can neglect the i(t)dLvar(Ψ(t))/dt) item of the equation (1.3) due to its tendency to zero.

When an IM is supplied by an unbalanced voltage system then already fast changes in Ψ will overlay a slow component of Ψ.

A similar phenomenon takes place at the beginning of IM start-up and it is shown on the left side of Fig. (1.2).

Therefore the initial explanation with a help of equation (1.5) is more versatile.

Fig. (1.2))

Simulation results for the start-up of a three-phase IM of type AZHV250M2RUHL2 supplied by a sinusoidal and balanced voltage system.

Parameters and characteristics of 110 kW IM are given in Table 1.1.

Table 1.1 Parameters and characteristics of 110 kW IM, type AZHV250M2RUHL2.

Fig. (1.3))

Simulation results for the operation of a three-phase IM of type AZHV250M2RUHL2 supplied by a non-sinusoidal and unbalanced voltage system.

The total moment of inertia of the rotating masses, reduced to the rotor shaft, used in the simulation is J = 2.43 kg.m². An approximation of the Lμ(ΨμƩ) function for IM of type AZHV250M2RUHL2 is shown in Fig. (1.4).

The rated operating mode of AZHV250M2RUHL2 corresponds to the value of ΨμƩ = 0.815 Wb.

Magnetic flux ΨμƩ = 0.815 Wb corresponds to the rated/nominal operating mode of AZHV250M2RUHL2.

The graph in Fig. (1.4) consists of several combined lines due to the fact that a single TABLE block from OrCAD [10, 11] was used to define this function, which sets the relationship between the input and the output signals by five pairs of values. To get a smoother shape of the graphs, one can use a parallel connection of several TABLE blocks.

Fig. (1.4))

Approximation of the vs curve for IM of type AZHV250M2RUHL2.

CONCLUDING REMARKS

If the assumption of similarity between the change in the absolute value of current and variable component of the inductance is true, then it is appropriate to use the nonlinear coefficients with the inductances in ODEs of electromagnetic devices. In these inductances, the effect of magnetic saturation is substantiated and represents functions of inductance vs current or magnetic flux