This article is dedicated to a burning problem of classification of phase portrait of a set of three simultaneous linear differential equations rendered in two-dimensional space R2. The suggested classification is a generalization of the existing researches and gives the possibility to distinctly outline dynamic tendencies of the third order linear patterns under the conditions of bifurcation inherent for automatic control systems (ACS) with changeable structure. The analysis of phase paths of the third-order linear system which is described by nondegenerated system of differential equations around single regular balance point M(0;0;0) and also phase paths with composite points of codimension 1 and codimension 2. It has been shown that the paths appearance depends upon the structure of characteristic matrix of the system of its own figures and the projections of phase portraits in phase space R3 on specific surfaces correspond to phase portraits of linear system of other order. Within this framework, phase portraits of the third-order linear system have been assigned names. The system of nondegenerated type of balance points with corresponding analytical conditions of origin and names is represented in a tabulated form. Special focus was made on the case of zero characteristic matrix which corresponds to phase portrait consisting of stationary point only. Application of the suggested classification system has been considered by the example of the automatic control system of the voltage of the DC generator. The sequence of actions and conclusion as per dynamic properties of ACS based on its phase portrait has been shown.

Analysis of Phase Trajectories of the Third - Order Dynamic Objects

The paper analyzes negative impact of poor-quality electricity on the performance of electrical installations. Particular attention is paid to the reasons for the deterioration of the quality of electrical energy in the power supply systems of industrial enterprises since it leads to a decrease in the reliability and efficiency of electrical consumers. Different character modes of consumption and generation of reactive power in the lines with 27.5, 10 and 0.4 kV have been determined. A wide variation in the statistical characteristics of power quality indicators and significant distortion curves current form have been determined that will help elaborate more accurate methodology of measures to improve the quality of electricity networks of non-traction consumers.

Evaluating the Effect of Electric Power Quality upon the Efficiency of Electric Power Consumption

Three-phase asynchronous motor with a short-circuited rotor has been described with the help of a model convenient for the analysis of changes of its power indices in terms of different values of all the indices of supply voltage quality. Mathematical model of electromechanical converter with the power of 11 kW has been verified based upon the results of industrial experiment in the workshop of “Ukrspetsservis” Ltd. While checking the validity of the model, the obtained values of relative mean square error of modeling make it possible to be used for the purpose of computational studies of asynchronous electric motor energy efficiency.

Development and Verification of Dynamic Electromagnetic Model of Asynchronous Motor Operating in Terms of Poor-Quality Electric Power

The article is dedicated to the regulation of regulation of the speed of the electric drive. It is noted that the main way of imparting astatic properties to an automatic system is to use integrating corrective devices-regulators. The phenomenon of parametric astatism is established and the conditions for its occurrence are found. The phenomenon of parametric astatism in a complex electromechanical system of speed control for an AC electric drive with an asynchronous motor and vector control is considered. It is shown that parametric astatism takes place in two-mass electric drives of direct and alternating current with a constant and variable as a function of the load speed, provided that the control part of the system is synthesized using root methods. This applies to systems of modal control and systems where the synthesis of low-order rate controllers is based on the polynomial method. It is determined that the majority of the standard distributions considered above allow one to obtain parametric astatism in the system.

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On peculiarities of regulation of electric drive with two-mass mechanical part and non-linear load

The problem of the increased potential of the rail-ground with the flow of electric trains reverse traction current operating in the area of the AC feeder is considered. The studies were caused by the need to ensure the electrical safety of the staff and the infrastructure facilities reliability when passing trains of increased weight. In the process of experimental studies of the high potentials causes in rail circuits, data were obtained indicating the complex nature of the electromagnetic interaction of reverse current harmonics with rail circuits. For studies of electromagnetic compatibility between railway subsystems, it requires the use of simulation programs that help to reveal most critical conditions for electromagnetic compatibility conditions, and allow the evaluation of electromagnetic interference from rolling stock in rails in the worst conditions. A developed the traction current harmonics distribution model in the rails is considered in this paper. The traction current harmonics distribution in the rails was calculated for a direct AC power supply of 25 kV depending on the distance from the power supply substation, the conductivity of the railway and the ground and the number of trains in the feeder zone. The results of the harmonics distribution simulation are in satisfactory agreement with the experimental data and will allow the technical measures development to reduce the rail-ground potential.

Impact of Return Traction Current Harmonics on the Value of the Potential of the Rail Ground for the AC Power Supply System

Track circuits are of fundamental importance for the safety in railway systems. Thus, they must be immune from electromagnetic interference (EMI). To this aim they must be accurately characterized. The paper deals with the determination of spectrum composition of traction current with the electric traction. A method of measuring the parameters of track circuits (time and amplitude parameters of code current, flowing in rail lines, an input impedance of a track circuit, characteristic impedance and propagation constant) is considered Automated method of measurement of parameters of track circuits and harmonics of return traction current was elaborated by the car-laboratory. The harmonics coinciding with the code frequency are inadmissible for the track circuits.

About Electromagnetic Compatibility of Track Circuits with the Traction Supply System of Railway

An investigation of propagation of return traction current harmonics in rail lines was carried out for railway sections with d. c. and a. c. traction. A mathematic model of traction supply net was proposed. The improved mathematical model of traction supply net allows us to estimate the propagation of harmonics and traction return current along the feeder zone with different kind of power supply system. The impedance of a power network was determined for low frequency interferences of return traction current. Also this paper deals with the determination of spectrum composition of traction current and most dangerous harmonics for track circuits.

Propagation of Harmonics of Return Traction Current in Rail lines

The diagnostic tool of code current parameters and return traction current interferences is proposed. This apparatuses allows to analyze the spectrum and the value of interferences in the track circuits and to determine their sources. The research results are illustrated. The probability characteristics of return traction current interferences are determined.

Research on Return Traction Current Harmonics

This paper deals with the research and comparative analyses of traction power system and electromagnetic interferences in the return traction current on the Railways of Ukraine, Europe and United Kingdom. The spectral compositions of return traction current were evaluated experimentally.

Analysis of Electrified Systems and Electromagnetic Interferences on the Railways

The questions of propagation of traction current harmonics along the length on an homogeneous feeder zone at the one- and two-end power supply and the definition of their influence on code current flowing in rail lines were elaborated. The mathematic modelling of a traction supply net was carried out. The spectrum of traction current and the most dangerous harmonics influencing the track circuits were assessed for the application of electric locomotive with the PWM-invertors. To this end, the values of current and voltage on the primary winding of the traction transformer of the double-section electric locomotive were determined.

Kseniia Serdiuk

Papers

About Electromagnetic Compatibility of Track Circuits with the Traction Supply System of Railway

Propagation of Harmonics of Return Traction Current in Rail lines

Research on Return Traction Current Harmonics

Analysis of Electrified Systems and Electromagnetic Interferences on the Railways

Mathematic Modelling of Distribution of Traction Current Harmonics