Transient analysis on track circuits in crack state

The transient transformation of track circuit between the normal state and crack state is analyzed. Firstly, the model of crack state of track circuit in the complex frequency domain is established. According to the boundary conditions of crack state of track circuit, the expression of current traveling wave in the receiving end is obtained in complex frequency domain. Based on it, the numerical solution in the time domain is got by FFT&IFFT conjunction with quotient-difference (Q-D) algorithm. Then, the steady state of circuit in normal state is taken as state of zero moment of track circuit, the transient process from normal state transform into crack state is solved. Thirdly, the Electro-Magnetic Transient Program (EMTP) model of crack operation is established to prove that the transient analysis result is correct. It provides a power theoretical basis to distinguish the operation state of track circuit by transient analysis.


Introduction
In order to analyze transient process of track circuits in crack state, there are three problems will be solved.The first is how to solve the transmission line equation, the second is how to get boundary conditions according to rail fracture state equivalent model, the three is that crack state complete response analysis with the initial condition of steady values in normal state.
The solution of distortion transmission lines was solved by analytical method in [4].However, this method is not suitable to track circuit, as the track circuit belongs to distortion transmission lines.The solution of track circuit transmission lines equation is difficultly transformed by inverse Fourier transform (IFFT) or inverse Laplace transform(ILT),so it is difficult to solve track circuit transmission line equation by analytical method.The track circuit is analyzed under specific loads in time domain in [5], but the analytical solutions has no general.The transmission line equation is solved by numerical method in most literatures.Numerical analysis have some weaknesses.For example, the order number of transfer function is higher, the computation complexity is bigger, and the problem is simplified relatively complicated [6][7][8][9].FFT&IFFT conjunction with quotient-difference(Q-D) algorithm does not refer the zeros and poles of transfer function ， and the computation complexity is lower [10].When the steady state of track circuit in normal state is taken as state of zero moment of track circuit in crack state, the transmission line is four order linearity non homogeneous partial differential equation.Based on the solution of high order linearity non homogeneous partial differential equation, fourth order indefinite integral which is difficult to solve will appear [11,12].

The crack states model of track circuit
The equivalent circuit model of track circuit crack state in complex frequency domain is shown in Fig. 1.
Where: ( ) s usis the sending end voltage source of track circuit, ( ) Zs is internal impedance of track Where: ( ) s us is the sending end voltage source of track circuit, ( ) Zs is equivalent impedance of the crack point.

The crack states expression of track circuit travelling wave
According to reference [5], in the initial condition of zero, the expression of voltage and current of each rail to ground can be written to (1)-( 4): (1) Where: ( ) ( ) ZB s is the characteristic impedance between two of rail, ( ) is the characteristic impedance from arbitrary rail to ground.( )


is propagation constant of between two rail.The integral coefficient of the section 1 l can be described by ( ) ( ) The integral coefficient of the section 2 l can be described by H s H s .
( ) ( ) H s H s are obtained by using the boundary conditions of track circuit in crack state.According to the condition1~condition5, (5)~( 12) can be got.Condition1:when 0

ZB s ZB s H s H s H s H s ZB s ZB s ZB s K s K s K s K s H s H s H s H s Zs
( ) ns is defined as ( ) ns is defined as ( ) , ( ) . ( 13),( 14) can be obtained according to ( 5)~ ( 12)  ，in arbitrary point of track circuit, the expression of voltage ( ) , u x s and current ( ) are obtained in complex frequency domain: , 2 e 2 e Transmission line whole response is the sum of zero state response with zero input response, zero input response can be calculated according to the principle that energy storage component of charging and discharging is reciprocal.Zero input response is discharge process, which the steady-state value of the previous working is regard as the initial values of current work.So, the zero input response can be regarded that zero state response minus the steady state of track circuit [13,14].
We assume the rail crack when track circuit energized after 10ms , ( ) ( ) In order to validate the result of theoretical analysis, the EMTP model is set as figure3 and the simulation result is shown as figure4 [15].

Conclusion
The equivalent circuit model of track circuit crack states in complex frequency domain was established.Based on it, the expression of current (voltage) travelling wave of track circuit in crack operation was obtained in complex frequency, The numerical solution was solved by FFT&IFFT conjunction with quotient-difference (Q-D) algorithm.Simulation result show that the method related in this paper can accurately analyze the crack operation of track circuit, and provide a novel method to realize transient analysis of track circuit.

Figure. 1 .
Figure.1.The equivalent circuit model of track circuit crack states in complex frequency domain

b
Zsis defined as the equivalent impedance of the crack point in crack model of track circuit.When ( ) b 0 Zs → , the track circuit operates in normal state, when ( ) b Zs = , the track circuit operates in broken state, when ( ) ( ) b 0, Zs  , the track circuit operates in crack state.