The internal parameter of UNI 60 rail is calculated by using finite elements methods. Steel's characterizations by its normal magnetization curve and by complex magnetic permeability are here considered and included into the proposed FEM models. Rail's resistance and internal inductance in function of current and frequency are calculated using both FEM and analytical models. The results obtained at the frequency of 50 Hz are compared with few measurements available, and then they are extended to other frequencies.
The railway electric system is usually modeled by a Multiconductor Transmission Line (MTL) [
Rails are one of the most important elements in the railway system; they are both guide of the railway vehicle and conductors of the electric system. In fact, rails are the common conductors between traction and signalling circuits and the interface between these circuits and the ground.
Their shape and the steel’s properties are chosen taking only mechanical requirements into account; electric and magnetic characteristics of the rails are a consequence of these primary choices. As a result, rail cross-section is far from the circular shape and the steel’s magnetic behaviour is nonlinear and hysteretic. The electric and magnetic characteristics are also neither complete nor certain; generally, they are derived from empirical formulas and they are based on the little data available in the literature. The computation of the tracks’ representative parameters is, at the same time, a fundamental and an extremely critical task in the calculation of the total system’s parameters. It is important to note that the simulation of the railway electric system requires a rail large signal model.
The main problems in the calculation of the parameters associated with rails are related to the rail’s cross-sectional geometry and to the magnetic behaviour of the steel.
As a matter of fact, the complexity of the cross-section does not allow an analytic solution of the magnetic field and current density inside the rail, even in presence of material with linear magnetic and electric behaviour. The accurate calculation of the rail’s internal parameters must be carried out using numerical techniques; the rail’s representation with an equivalent conductor with a circular cross-section must be considered the simplest, but the heaviest, approximation.
Furthermore, the nonlinear behaviour and hysteretic phenomena in the steel make the internal parameters of the rail dependent on both the frequency and the current. In order to include the rails into an MTL large signal model it is necessary to introduce a proper linearization in the steel’s magnetic behaviour.
Some papers try to address the problems underlines. In particular, in [
In this paper, the methods proposed in the literature for the representation of the steel’s magnetic permeability are introduced and successively applied to the calculation of the internal parameters for large signal simulation, with reference to a UNI 60 rail, by using FEM. In particular, two different FEM rail’s models are proposed. These models are based on two different kinds of linearization of the magnetic material behaviour. The first one takes into account the normal magnetization curve of the material and adopt the linearization method based on the sinusoid equivalent concept. The second rail’s FEM model includes both the effects of hysteresis and saturation by using the linearization based on complex magnetic permeability function.
The obtained results are also compared with experimentally measured data [
The internal impedance of a conductor is related to the distribution of the magnetic field and current density on the conductor’s cross-section; these distributions depend both on the shape of the cross-section and on its electric and magnetic characteristics.
For a typical geometry with a semi-infinite slab of isotropic and linear conductive material in which sinusoidal current flows, the modulus of the magnetic field and current density fall down exponentially with the increase in the distance
In a ferromagnetic material, both saturation and hysteresis are observed in the magnetic behaviour. These phenomena have been measured in a sample of rail steel [
In the next part of this chapter, some techniques for deriving an approximated value of the magnetic permeability, starting from the hysteresis loop and/or the normal magnetization curve, will be introduced. These techniques allow the deduction of a permeability value that takes into account the quantitative aspects related to presence of saturation and/or hysteresis in the material’s magnetic behaviour.
The effect of saturation on the magnetic field and current density distribution can be evaluated with reference to the typical geometry (semi-infinite slab) previously described. In this type of analysis the material magnetic behaviour is represented with an appropriate function that relates the induction
Steel is both nonlinear and hysteretic; in order to take into account only the saturation effects the normal magnetization curve is introduced into this model. Numerical results are here obtained taking into account this curve measured on a rail’s steel sample in [
The distribution of magnetic field in the semi-infinite slab conductor previously introduced can be computed by solving the following differential equation:
Saturation modifies the attenuation of the magnetic field inside the conductor with respect to the linear case and it introduces distortion in the waveform. The reduction of the magnetic field’s maximum amplitude with the increase in distance from the surface is, however, very close to the linear case exponential decay. Figure
Magnetic field within a semi-infinite slab with nonlinear material.
The rapidity with which the amplitude of the magnetic field falls down depends both on the frequency and the maximum value of the magnetic field on the surface of the conductor. The effective skin depth can be defined as the distance from the surface where the magnetic field maximum value is reduced by a factor of
It is important to underline that effective permeability does not depend on the frequency; it depends only on the magnetic field strength on the surface of the slab. This is true only if the non-linear BH function that represents the magnetic characteristic does not change with frequency. Figure
Effective skin depth at 50 Hz and effective permeability.
Real and imaginary components of the complex magnetic permeability.
It is known that the hysteresis phenomenon causes time delay between the induction and the magnetic field, and that the loop area represents the energy converted in heat in a cycle. The shape of the hysteresis loop for a linear hysteretic material is an ellipse, in fact induction and magnetic field pulse at the same frequency and the angle between them can be represented, in the phasor domain, considering a complex value of magnetic permeability
Referring to the semi-infinite slab previously described, the skin depth takes the following complex value
In the case of the simultaneous presence of saturation and hysteresis, the behaviour of the material can be approximated using an opportune value of complex magnetic permeability evaluated as follows [
The angle of complex permeability can be determined from the energy analysis of the fields
The angle of the complex permeability depends on the maximum value of the magnetic field, which is related to the shape of the hysteresis cycle. The angle of complex permeability can be considered independent of the frequency until the shape of the hysteresis cycle does not change with frequency. The case of small hysteresis losses corresponds with little values of the complex magnetic permeability angle.
The evaluation of the modulus of complex magnetic permeability depends on the distribution of the fields into the material. In the condition of small hysteresis losses the attenuation is reasonably similar to the only saturated case described in Section
The components of complex magnetic permeability defined in (
For conductors with cross-section different from the circular shape it is possible to apply relation (
The substitution of the rail with a circular equivalent conductor avoids the problem of calculating the fields into the cross-sectional surface, that is related to the complexity of the geometry, and it requires the setting of a proper rail equivalent radius. The internal impedance of a circular cross-section conductor is defined by the following relation:
The equivalent radius can be defined by imposing the equality, at a given frequency, between the actual value of one of the two internal impedance components and the same value calculated by the equivalent circular model. Generally, DC and AC high frequency resistance is the actual values used to define, respectively, the DC and AC high frequency equivalent radius. The calibration of this model is very simple and it do not require measurement or simulations on the actual conductor, but the regulation can be done only at one frequency and it is not possible to regulate both the resistance and internal inductance simultaneously. As a consequence, it is impossible to fit the real frequency dependant internal impedance components in a wide frequency range and model’s results are accurate only for one component and in a small frequency range.
In order to achieve a more precise analytical model, some authors propose a 4 parameters tubular equivalent conductor [
For DC calculation, two different values for the equivalent radius can be taken into account. The first one is the radius that gives the same rail area and it produces the same DC rail resistance
The second one is based on the geometric mean radius. The geometric mean distance (GMD) and geometric mean radius (GMR) of an irregular conductor are needed for inductance calculation under steady-state condition [
As frequency goes up they the nonuniform distribution of the current density caused by the skin effect becomes relevant and these methods cannot be applied. Due to the presence of ferromagnetic material and the dimension of the rail, the upper limit of the very low frequency band is about a few Hertz. It is important to note that even at 1 Hz, a 100 A current produces a straightforward non-uniformity in the current density.
In AC high frequency the skin effect forces the current in the rail periphery. In this case a equivalent perimeter radius must be taken into account:
In the transition range, included between the low and high frequency range boundaries, an intermediate equivalent radius must be taken into account and the area and perimeter equivalent radius represent the extreme values for the rail equivalent radius.
It is important to note that transition range include power frequency and the lower part of the harmonic spectra produced by traction converters. The definition of this frequency boundaries is however difficult, and considerations about the conductor’s shape and dimensions, and the skin depth will be helpful for this task. Finally, these limits depend on the current due to the complex magnetic permeability function. However it is possible to consider about three decades between these two limits.
The calculation of the internal parameters of a rail can be carried out also using finite element softwares. Such tools resolve the magneto-electric problem in a numerical way starting from the geometry of the problem, the characteristics of the materials, the sources, and the boundary conditions.
Thanks to the uniform cross-section of the rail, the model can be performed in 2D on the cross-section surface, reducing consequently the simulations computational time without lack of precision. The model has been implemented using Ansoft Maxwell version 10. A rail type UNI 60 is represented in the vacuum in order to neglect any kind of proximity effects; this choice matches the conditions under which the circular equivalent conductor model is derived. Moreover, the measurements conditions in [
The simulations are performed using “Eddy Current Field Solver.” This solver assumes that all currents are sinusoids oscillating at the same frequency, and they produce a time-varying magnetic field in the plane perpendicular to the conductors in which currents flow. The magnetic field induces eddy currents in the source conductor and in any other parallel conductors. Eddy Current Field Solver uses FEM to compute the magnetic vector potential and the electric scalar potential in the phasor domain using the following relations:
In a 2D problem, induction B lies in the
The simulations are performed imposing the current on the rail’s surface and taking into account that the empty space extends infinitely out, placing “balloon” condition on the boundary of the model.
The average internal energy and the average losses per unit length are calculated by integrating the average values of the specific correspondent quantities over the surface of the conductor obtained by FEM simulations. Under sinusoidal conditions, these quantities are given by
The internal impedance components per unit length are determined from a port-process operation that consider the magnetic stored energy (
The FEM overcomes one of the main problems in the calculation of the parameters: the one associated to the rail’s cross-sectional geometry. The problem linked to the magnetic behaviour of the steel can be faced developing two different FEM models based on two different kinds of linearization of the magnetic material behaviour.
The first one takes into account the first magnetization curve of the material and adopts the linearization method based on the sinusoid equivalent concept. Eddy current nonlinear solver calculates the effective permeability of nonlinear materials as the ratio of the effective induction
These quantities are defined starting from energy considerations. In fact, assuming a sinusoidal magnetic field with amplitude
In the same way, it is possible to define the
In this way it is possible to calculate internal parameters taking into account only the effects of saturation in an approximated way.
Figures
Rail resistance calculated by using FEM nonlinear solver.
Rail internal inductance calculated by using FEM nonlinear solver.
In fact, as in this software it is not possible to define the hysteresis loop in the material properties, the magnetic losses due to a time varying field and the effects of saturation can be included in the model only using a proper value of complex magnetic permeability for each value of current.
The complex magnetic permeability function is calculated by (
Current density inside a rail and flux lines computed with a current of 100 Armsand 1000 Armsfor different frequencies.
The use of the function of complex magnetic permeability represents a very efficient method to calculate the rail parameters using the finite element software and taking into account, in approximated way, both saturation and hysteresis losses. Figures
Current density inside a rail and flux lines computed with a current of 100 Arms and 1000 Arms for the typical railway power frequencies.
Rail resistance calculated by using FEM complex permeability.
Rail internal inductance calculated by using FEM complex permeability.
It is important to underline that both FEM models require the solution of the magneto-electric problem. For this reason, this kind of simulation needs more computation time than the equivalent circular conductor model. Finally, it is important to note that computation time of FEM models increases with the requested precision and with frequency because the number of triangles in the region near the rail’s perimeter increases with the skin depth reduction.
The graphs that show the dependence of internal parameters on the effective current and the frequency obtained from the two FEM proposed models are quite similar in shape, but the difference in values is remarkable.
The comparison between models’ results and measured internal parameters of a UNI 60 rail type can be carried out only at the frequency of 50 Hz [
Resistance versus current at 50 Hz.
Internal inductance versus current at 50 Hz.
The resistance calculated with the nonlinear FEM model is less than the resistance calculated with the linear FEM model with ferromagnetic material characterized by complex permeability; the internal inductance calculated with the nonlinear FEM model is bigger than the internal inductance calculated with the FEM model based on complex magnetic permeability. Such differences are associated with the hysteresis losses that are neglected in the nonlinear FEM model based on the normal magnetization curve. In fact the effects of hysteresis are to increase the AC resistance and decrease the internal inductance by reducing the average energy stored in the material [
Resistance and internal inductance increases with current. The magnetic permeability increase with current from 0 A to about 800 A, and only above this current saturation will be noticeable.
In order to achieve an expression between internal parameters and the current at the frequency of 50 Hz, an interpolation of measured data has been computed. The resistance and internal inductance have been interpolated using third- and fourth-order polynomials, respectively [
The FEM model based on the normal magnetization curve is more sensitive to the accuracy of the parameters than the model based on the complex magnetic permeability. Small changes in the normal magnetization curve, especially in its initial values, will produce large changes in the internal parameter computed using this model. Small changes in the shape of hysteresis loops will produce small changes in the complex magnetic permeability function.
Although the characterization of the rail steel is not available with reasonable precision, the results obtained with the FEM models are very close to the measured values. The model that gives better results, both in resistance and internal inductance, is the FEM one based on the complex magnetic permeability. The little difference between the internal parameters curves calculated using this model and the measured data interpolation one can be associated to the uncertainty both in the parameters used in the models and in the measured data.
In Table
Error between measured data and FEM.
FEM complex | FEM nonlinear | |||
permeability | solver | |||
Current | ||||
100 | −18,2 | −9,0 | ||
200 | −15,7 | −19,6 | ||
300 | −11,6 | −26,5 | ||
400 | −7,8 | −30,4 | ||
500 | −5,5 | −32,4 | ||
600 | −2,5 | −32,0 | ||
700 | −28,1 | |||
800 | −22,0 | |||
900 | −2,5 | −15,8 | ||
1000 | −6,7 | −11,0 |
Two methods based on the FEM software for the calculation of the internal parameters of rail with reference to the rail type UNI 60 have been introduced. The obtained results have been compared to data available in the literature. This comparison shows that the accurate evaluation of the internal parameter of a rail can be performed taking into account the rail geometry and hysteresis effect using FEM software.
It is important to emphasize that the results are related to the precision with which the magnetic and electric characteristics of the rail steel are known [
The results here obtained depend on the shape of the hysteresis cycle. In order to achieve good accuracy, it is necessary to know exactly the hysteresis cycles’ family for various values of excitation and frequencies.