ABOUT TRACK CIRCUIT CALCULATION METHOD DEPENDENT ON FERROMAGNET PROPERTIES IN CONDITIONS OF TRACTION CURRENT NOISE INFLUENCE

Purpose. The work is intended to investigate the electromagnetic processes in impedance bond in order to improve noise immunity of track circuits (TC) for safe railway operation. Methodology. To achieve this purpose the methods of scientific analysis, mathematical modelling, experimental study, a large-scale simulation were used. Findings. The work examined the interference affecting the normal performance of track circuits. To a large extent, part of track circuit damages account for failures in track circuit equipment. Track circuit equipment is connected directly to the track line susceptible to traction current interference, which causes changes in its electrical characteristics and electromagnetic properties. Normal operability, performance of the main operating modes of the track circuit is determined by previous calculation of its performance and compilation of regulatory tables. The classical method for determination of track circuit parameters was analysed. The classical calculation method assumes representation of individual sections of the electrical track circuit using the quadripole network with known coefficients, usually in the A-form. Determining the coefficients of linear element circuit creates no metrological or mathematical difficulties. However, in circuits containing nonlinear ferromagnets (FM), obtaining the coefficients on the entire induction change range in the cores is quite a difficult task because the classical methods of idling (I) and short circuit (SC) are not acceptable. This leads to complicated methods for determining both the module and the arguments of quadripole network coefficients. Instead of the classical method, the work proposed the method for calculating the track circuit dependent on nonlinear properties of ferromagnets. Originality. The article examines a new approach to the calculation of TC taking into account the losses in ferromagnets (FM), without determination of equivalent circuit quadripole network coefficients. When building the FM reversal model in parallel magnetic fields, the most accurate methods are the approximation ones that take into account not only the changes in values and over time, but also their derivatives. The development of computer hardware and software makes real the mathematical methods for calculating TC with significant change in ferromagnetic inductance, including the saturation areas. Herewith, it is important to search for approximating analytical expression that describes the dynamic limit hysteresis loop (HL). Practical value. The changes in the electrical parameters of the same TC were analysed using the classical and the new calculation methods, the difference made less than 10%. The work introduced some measures to increase operational noise immunity of TC.


Introduction
Experience has shown that certain conditions and operating modes of the power network create a powerful influence of the traction current noise on the work of electric track circuits (TC).Such exposure results in magnetization of nonlinear ferromagnets (transformers, impedance bonds and components containing ferromagnetic core) in track circuits and, consequently, in failure of TC and signalling systems.Adverse conditions [11,12,13] for TC operation are created in the following cases: 1) At a certain switching circuit of DC traction network of splicing stations the level of permanent traction current potentials «rail-to-earth» in the rail network increases.It results in increased leakage currents into the open-line track of AC electric traction lines, causing magnetization of impedance bonds and track transformers, and disruption of short (less than 500 m) track circuits; 2) In the areas of AC electrified railroads the ice on the contact network collectors creates an electric arc, accompanied by electrical transients in the power circuit of an electric locomotive.Con-stant component of this current at the rail line asymmetry also causes saturation of ferromagnets; 3) Switching on the locomotive main switch in a certain voltage phase of the circuit power line also leads to transition process with the consequences given above; 4) When a laden electric locomotive enters the neutral section on the current collectors there are several electric arcs repeated aperiodically; this leads, as mentioned above, to the occurrence of transients and failures of TC.From the above it follows that improvement of track circuit noise immunity in the signalling systems in conditions of permanent reverse traction current influence is an actual scientific and technical problem.

Purpose
The purpose of this work is to examine the causes of traction current harmful influence on TC operation, to create the calculation method for TC with nonlinear ferromagnets (FM) and to propose technical solutions to improve the reliability of TC and railway automation systems.

Methodology
Disruption of ТC operation.Let's consider one of the causes of TC operation disruption -longitudinal asymmetry of track lines.Measurements show that the asymmetry coefficient is higher in winter than in summer due to increased insulation resistance.It is established that the best way to approximate the conductive joint resistance distribution is asymmetric Weibull law, resistance variation reaches 1.8 … 15 m.e.r.(meters of entire rail); longitudinal asymmetry coefficient ranges from 10% to 12% -in summer, up to 18% -in winter [1,6].

Fig. 1. Equivalent circuit of track lines
The EMF equation for equivalent circuit contour of the track (Figure 1) consisting of two rails, equipment impedance `igo Z , ire Z and choke cables, for the difference in currents can result in the following expression: where EL I -electric locomotive current; Z perform the symmetric action, but their impedance for the signal current under the track integrity monitoring conditions is small and can be neglected.
For DC the equation ( 1) can be written as: Thus, the longitudinal asymmetry coefficient of track line is: where RC R ∆ -difference in track resistance due to resistance of connectors.
In short TC (less than 500 m) longitudinal asymmetry is one of the main causes of traction current asymmetry Measurements show that the resistance of the steel DC rail connector is distributed among its components, so [4,3]: Rules for protection of metallic structures from stray-current corrosion allow increasing rail resistance through the joints by max 20%, while the DC asymmetry coefficient is Approximate calculation of asymmetry for alternate traction current is performed with the following assumptions [3,8]: − Resistance of rails differ among themselves by the amount of resistance of connectors; − Connector losses do not depend on the frequency; − Resistance in points of welding and junctions cable -nipple -plug -rail depends on the frequency as much as the internal resistance of the rails.
− Herewith, the minimum coefficient of longitudinal asymmetry: where i Z -internal resistance of rails, 0.35е j35 Ohm/km; rail loop resistance p R = 0.1 Ohm; Z = 0.8e j65 Ohm/km -rail resistance to 50 Hz current.For higher frequency the coefficient Z K is re- duced to 1.608% for 420 Hz frequency and 1.376% for 780 Hz.
Let's assess the numerator of the fraction (3) for TC that most often fail with length of 0.25 … 0.5 km.We assume that Let us consider the proposed TC calculation method for nonlinear FM modes of TC circuits, based on the mathematical model of ferromagnet reversal magnetization [2,4].

Originality and practical value
Mathematical description of model.We assume that ferromagnetic saturation leads to changes in induction (intensity) of the magnetic field of DT.1.150core according to the complex law ( ) ( ) where k = 1, 2, …, m , nd measuring of quadri- pole network using the known methods of idling and short circuit is too complicated [2].
It is known that when there is cyclical change in the external magnetic field the ferromagnetic induction retards in phase from the tension and enables the cycle of dynamic hysteresis loop (HL) (Fig. 2 (a)).HL area defined the power losses caused by reversal magnetization, vortical currents and magnetic viscosity [3].
The mathematical model that accurately describes the research HL is as follows: bm Ae −γ -makes sense when building the specific loop exits and depends on constructive features of the subject) and allows taking into account the influence of harmonic spectrum for ferromagnet reversal input current.The values β -approximation coefficients of the function ( 6), are determined using numerical methods by HL experimental end value and magnetization curve, taken at the maximum values of induction and signal frequency.The first term ( 6) describes the basic magnetization curve (BMC), the second one describes dissipative processes both for hysteresis and vortical currents, and depends on the magnetic viscosity.( ) ( ) ( ) We present the hyperbolic functions in ( 7) by rows: and keeping harmonics no higher than the second one, after transformations we obtain: ( ) ( ) ( ) ( ) We examine the term 0 h in the formula (8), since this is the component of the magnetic field strength that leads to the choke core saturation and, ultimately, to failure of TC receiver.
When applying the numerical estimate for field intensity dependence on induction in the formulas ( 6) and ( 8), the following parameters are used: The impedance argument 0 Z can be determined by the value of power losses for vortical currents and hysteresis [5] From (10) and the equation of magnetized winding circuit of the choke DT.1 with the resistance r : we obtain EMF and phase for the first harmonic: 1 0 sin( ) If induction where ), r µ -magnetic perme- ability of steel is taken 500, d = 0.055m, b = 0.18m -thickness and width of the package of core plates [6].Calculation of complex impedance growth of DT.1.150core winding due to vertical current effect is possible thanks to (13): (1 ) 0, 2 where П S a b = ⋅ -flowed cross-section area of the package; a = 0.175m m -steel package height.
For low frequencies (below 900 Hz) Let us consider the peculiarities of calculation of the track circuit with nonlinear ferromagnets.

Findings
Calculation of TC operation modes are conducted using the structural equivalent circuit (Fig. 3) of impedance bond, presented by the circuit of Lshaped quadripole network and ideal transformer.
Determination of quadripole network coefficients (usually in A-form) on the entire change range in the induction cores, which exceeds the saturation limit, is quite a difficult task because the classical methods of idling and short circuit are not acceptable.The proposed method of TC calculation facilitates the solution of the above problem.The calculations require the FM hysteresis loop (Fig. 2  The design equivalent circuit of choke excludes the leak inductance of windings and the given additional winding resistance that affects the calculation accuracy to a small extent. The calculation results for phase-sensitive twin TC at 25 Hz using the coefficients (15) with choke bias at the ends of TC showed coincidence with those obtained by conventional methods.
Table 1 shows comparative design parameters for the same TC, obtained by two calculation methods -classic and new, using a mathematical model of ferromagnet reversal magnetization.
Figure 4 shows the dependence of voltage at the receiver input resistance on the impedance module 0 Z .It can be seen that the end value of resistance at saturation of both chokes reaches 0.44 Ohms, which is 1.6 times less than the nominal value.Switch of the receiver to zero state is possible at saturation of one choke up to resistance 0 Z = 0.2 Ohm (Fig. 4).The first solution is to use DT.06.500S type, used for splicing stations, at the stations of impedance bonds with air gap [2,3].
The second solution is to use switching units [9], which excludes spreading of traction DC out of the splicing station rail network towards the range of electric traction AC that, beside IB saturation, eliminates electric corrosion of supports and metal elements at a distance of 30 km from the source of noise.The device is tested in the laboratory of DNUZT and in the field of Pyatikhatky splicing station of Prydniprovs'ka railway.
The third solution is to install symmetric resistive ( ) units [2,8] in short TC at AC electric traction.The impedance values of unit resistors are designed to exceed substantially the sum of all resistances in the rail line bridge circuit (Fig. 1).The unit resistors (Fig. 1) are switched on in series with choke cables at one end of the rail line, which is much cheaper than in the case of choke replacement.
The conclusion should emphasize the following: it is found that the cause of TC failures is ferromagnetic core saturation under conditions of rail line asymmetry, herewith the asymmetry of direct current magnetizing FM is 4 … 5 times higher than the asymmetry of alternating current.
The new method of TC calculation using the mathematical model of ferromagnet reversal magnetization allowed obtaining the numerical evaluation of maximum allowable asymmetry currents and input impedances at the circuit ends, which cause failure of signalling system devices.
The work formulated the technical measures for improvement of noise immunity and TC protection under conditions of powerful traction current noise.

Coefficients 1 α and 1 β 2 bm
of anhysteretic component(6)  are determined by the selected points on the BMC curve, the values A and γ are determined by expressions force, A/m.Let us consider the ferromagnetic reversal process affected by induction core created ( ) by direct current field and signal current field of 25 Hz (Fig.2 (b)).We substi-tute this expression as an argument in the formula(6)  and turn it omitting the exponential factor 2 Ae −γ , which does not affect the quantitative result of the first term.As the argument b changes according to the periodic law, then the functions and can be represented by Fourier series[10], in which the coefficients in the trigonometric functions are Bessel functions of different orders of imaginary argument m jx .The result is:

Fig. 2 .
Fig. 2. Dependence of field intensity on induction at 25 Hz frequency: a − harmonic signal; b − harmonic signal with bias :

2 2
cross-sectional area of the magnetic core covered with windings; l = 0.58maverage length of the magnetic line for flows.Then (14) can be written as real and imaginary parts.As a result we will get the formula for inductance evaluan , n S -wire length and cross section.