A Study on Determination of Electrical Impedance and Frequency Characteristics of a Subway System Using Finite Element Analysis

The paper provides a description of the analysis of a subway system (track-in-tunnel) by using FEM analysis and comparing to classical analytical approaches by Carson, Pollaczek, Bickford and Tylavsky. Reviews of methods to determine self and mutual impedance for electrified railroads are provided. These methods include frequency response and are directly applicable to a three-rail track DC track system. The analytic impedance models are built on CarsonPollaczek–Bickford equations, adjusted by Tylavsky, for two situations: when the ground is perfectly insulated and when considering the earth return current. For the latter, the authors assume a return current path only through the tunnel concrete structure below the railway track support structure. The model is extended by considering the effects of the soil beneath under tunnel as a conductor. The solution of finite element method (FEM) applied for the determination of impedance for the three-rail track subway train configuration, modeled and examined, consists of computational analysis based upon minimizing the energy of electromagnetic field. The paper continues by examining the frequency effects on the track and system. The track/trolley model developed by Tylavsky was modified such that the trolley feeder is provided by the power rail and used to calculate the return current through the traction rails. The subway train, supplied with a rectified DC power, is subjected to a significant harmonic content, which may affect the signal and control circuits. Both experimental data and preliminary analytical and numerical calculations are presented, showing the variation of resistances and inductances of the running track with the current magnitude and frequency response. In the study, a large frequency range was considered (15Hz to 5000Hz) in order to cover all of the significant frequencies used for control and signal systems in common tracks configurations, and for which measurements have been carried. It is then shown that the power and signaling characteristics of the modeled system can predict the magnitude of the perturbation current for different values of frequency. The current density profile is illustrated for the case of a concrete tunnel structure in a subway application. The last section consists of a discussion regarding future developments and further work. SYMBOLS AND NOMENCLATURE f -frequency Hz ω -angular frequency rad/s z -internal impedance per unit length, Ω/m zii -self-impedance of rail i per unit length, Ω/m zij mutual-impedance of rail i with respect to rail j per unit length, Ω/m [V] -column vector representing the rails voltages with respect to the earth [I] -column vector of rail currents [Z] -track impedance matrix, Ω/m I -current, A μ0 -magnetic permeability of the free space, H/m μ -magnetic permeability of the ferromagnetic rail, H/m σ, σg -electrical conductivity for rail and ground, S/m δ -skin depth, m lii -self-inductance of one rail per unit length, H/m lii,int -internal self-inductance per unit length, H/m lii,ext -external self-inductance per unit length, H/m mij -mutual inductance of conductors i j per unit length, H/m a -equivalent radius of the circular conductor, m y -length of the wire or rail, m de -depth of the return conductor with respect to the ground plane, m θ -hysteresis angle, rad Rlf,Rhf -internal resistance at low frequency, respectively at high frequency, per unit length, Ω/m Xlf ,Xhf -internal reactance at low frequency, respectively high frequency per unit length, Ω/m zlf ,zhf -internal impedance at low frequency, respectively at high frequency per unit length, Ω/m Proceedings of the 2013 Joint Rail Conference JRC2013 April 15-18, 2013, Knoxville, Tennessee, USA 1 Copyright © 2013 by ASME zs,ext -external impedance of the ground return conductor, per unit length, Ω/m zm -mutual impedance per unit length, Ω/m α -complex skin depth function, m h,hk,hm-height of the wire or rail above the conductive plan, m d -depth of the interface between the ground layers, m dij -distance between the wires or rails, m I0,I1 -modified Bessel function of order zero respectively 1 J0 -zero order Bessel function of a first kind r -radius, m A -vector potential, Tm B -magnetic flux density, T