# Research on Calculation of Rail Self-Impedance of Track Circuit Based on Partial Element Equivalent Circuit

Authors: Bin Zhao; Guanghao Yu; Dong Wang; Jingning Ou; Lei Chen
Journal: Security and Communication Networks
 (2022)

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## Abstract

The self-impedance of the steel rail, as the signal transmission medium in the electrified railway track circuit, has a direct impact on the track circuit’s transmission performance. The finite element method is the most common method for calculating rail impedance, although it has a number of drawbacks, including a complicated model, a long computation time, and poor accuracy. The partial element equivalent circuit (PEEC) approach is used in this paper to provide a method for determining rail self-impedance. Firstly, the rail equivalent method is determined by analyzing the physical model of the rail. Considering the skin effect, the PEEC model of the rail is established. The internal impedance of the rail can be obtained by solving the equivalent circuit. The external impedance considering the influence of the earth is calculated by the Carson impedance calculation formula, which is processed by the segmented linear approximation method. The rail’s self-impedance is determined using the two procedures together. Finally, the actual measurement data verified the PEEC method to calculate the rail impedance. Compared with the finite element method (FEM), the calculation accuracy of the PEEC method is higher. The current frequency, the height of the rail from the ground, and the earth’s conductivity impact on the rail’s self-impedance are analyzed. The results show that the PEEC technique can be used to calculate the rail’s self-impedance and that the impact of current frequency, rail height above the ground, and ground conductivity on the rail’s self-impedance may be accurately represented. The self-impedance computation of the track circuit provides a theoretical basis.


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## Body

## 1. Introduction

Rails are commonly utilized by track circuits to communicate train control information and verify the occupancy and idleness of the track, which is the fundamental equipment in the train operation control system, in addition to supporting train operation [1, 2]. Rail impedance is an important electrical parameter in rail circuit design, which affects the transmission performance of the rail circuit [3]. Because of the intricate external electrical environment, the weak signal carried by the track circuit is prone to inaccuracy in the external interference and has an influence on train safety. The accurate value of rail impedance is the cornerstone of the track circuit analysis. Rail impedance is composed of rail self and mutual impedance; the calculation of rail self-impedance serves as the foundation for analyzing the transmission characteristics of the rail circuit.At present, the main methods for calculating the impedance of medium are Carson’s theory, FEM, and method of moments (MoM). Carson et al. established a theoretical approach for calculating overhead conductor impedance, which offered a theoretical foundation for researchers to investigate the overhead conductor impedance problem [4]. Deri et al. approximated the self and mutual impedance of overhead conductors by considering the effect of skin effect and using complex permeability [5]. In addition to giving a more accurate way to calculate rail self and mutual impedance than the conventional method, the kernel function segmented linear approximation method is used to solve an integrand of the Carson impedance equation [6, 7]. The rail self and mutual impedance are calculated by FEM [8, 9]. Compared with the measured data, it is verified that the FEM has high accuracy to calculate rail impedance. However, the FEM is more complex to model and requires larger computational resources. Using a finite element model of rail track, the semianalytical method is applied to calculate rail impedance [10]. This method has higher precision in a low-frequency band and has limitations in the high-frequency band. The equivalent surface impedance model is proposed based on MoM [11] and verified by arithmetic examples. A model was developed by Hill and Carpenter to calculate the internal impedance of rails that allows for the hysteresis effect of rails as well as to study the variation of internal resistance and internal inductance with the amplitude and frequency of rail current [12]. Zhu et al. calculated the rail AC internal impedance by equating the rail as a tubular conductor theory, which is in good agreement with the FEM [13–15]. Considering the special ferromagnetic properties of rails, the traction network impedance was calculated by parallel multiconductor transmission line theory [16]. Wu et al. calculate the frequency-varying impedance of rails by a FEM of incremental permeability in the transient state. Compared with the steady state, the AC resistance and internal inductance values of P60 rails are increased [17]. The influence law of parameters such as frequency, soil conductivity, steel burial depth, and steel sparsity on rail impedance is explored using finite element and circuit analysis methods [18].The PEEC is a method for solving integral equations in a medium that exists in small cells, based on the electric field integral equation. The equivalent circuit is established through the electrical connection between the partial elements [19–21]. The essence of the method is that the electromagnetic effect of each discrete unit electromagnetic structure transforms into circuit elements in the corresponding equivalent circuit through the electric field integral equation [22–24]. This paper adopts the PEEC method to model and calculate the internal impedance of electrified railroad rails. The external impedance of rails is calculated by Carson’s theoretical formula which is deal with the segmental linear approximation method with the influence of the earth.This paper is organized as follows. Section2 gives an overview of the partial element equivalent circuit method. Section 3 discusses the establishment of the PEEC model for steel rails. Section 4 discusses the algorithm feasibility verification. Section 5 expresses the self-impedance calculation and analysis of steel rails. Section 6 concludes the article.


## 2. Partial Element Equivalent Circuit Method

According to Maxwell’s equations, when an external excitation electric fieldE⇀0 is applied to a space medium, E⇀r⇀,t can be expressed as(1)E⇀r⇀,t=E⇀0r⇀,t+−∂A⇀r⇀,t∂t−∇Φr⇀,t.According toE⇀=J⇀/σ (σ is the conductivity), (1) can be written as(2)E⇀0r⇀,t=J⇀r⇀,tσ+∂A⇀r⇀,t∂t+∇Φr⇀,t,where E⇀0r⇀,t is the applied electric field’s contribution to the electric field strength at r⇀ at the moment t, A⇀r⇀,t is the vector magnetic potential, and Φr⇀,t is the scalar potential.In a homogeneous simple medium where there is a body current and a surface current and there is no free net charge in the body but only a surface free charge, the electromagnetic vector bits and scalar bits can be expressed as(3)A⇀r⇀,t≜μ4π∫V′⇀J⇀tr⇀,tRdV′+∫S⇀′J⇀str⇀,tRdS′,Φr⇀,t≜14πε∫S⇀′ρstr⇀,tRdS′,where μ is the magnetic permeability (μ=μrμ0), ε is the dielectric constant (ε=εrε0), densities of bulk current and surface current of a medium are J⇀t and J⇀st, respectively, ρst is the charge density on the surface, r⇀ and r⇀′ are the field and source vector diameters, respectively, R=r⇀−r⇀′ denotes the distance between the field and source, t is the observation moment at the field, td is the hysteresis moment of the electromagnetic source, td=t−R/c, and c is equal to the speed of light in vacuum.(4)μ0=4π×10−7Hm,ε0=8.854×10−12Fm,c=1μ0ε0≈3.0×108ms.When the mordant to be equated is discretized into small-size cells, we assume that it is dissected by volume intoNv orthogonal hexahedra, numbered 1 to Nv, and the surface is dissected into Ns rectangular slices, numbered Nv+1 to Ns. The vector magnetic potential and scalar potential of the whole mordant after discretization can be expressed as(5)A⇀r⇀,t≜∑j=1Nvμ4π∫V¯j′J⇀tjtdRdV′+∑j=Nv+1Nsμ4π∫S¯j′J⇀stjtdRdS′,Φr⇀,t≜∑j=1Ns∑k=1Nj14πε∫S¯jk′ρstjtdRdS′,∑j=1Nskj=Kq,where Kq is the total charge corresponding to the node number j.The electric field integral equation applies to the current elementVi:(6)∫V′⇀iE⇀0dV=∫V¯i′J⇀σ+∂A⇀∂t+∇ΦdV.By dividing the integral equation for the electric field by the dimensions of a cross sectionAi of the current element along the direction of the current, we get(7)I^iAi⋅∫V′⇀iE⇀0dV=I^iAi⋅∫V′⇀iJ⇀σ+∂A⇀∂t+∇ΦdV,I^iAi⋅∫V′⇀iE⇀0dV=I^iAi⋅∫V′⇀iJ⇀r⇀,tσdV+∑i=1Nvμ4πAi∫V⇀′i∫V′⇀jI^i⋅I^jRij∂Jtjtd∂tdV′dV+∑j=Nv+1Nsμ4πAi∫V⇀′i∫S⇀′JI^i⋅I^jRij∂Jstjtd∂tdS′dV+1Ai⋅∫V′⇀i∂Φr⇀,t∂lidV=I^iAi⋅∫V′⇀iJ⇀r⇀,tσdV+∑j=1Nvμ4πAiAj∫V⇀′i∫V′⇀jI^i⋅I^jRijdV′dV∂Itjtd∂t+∑j=Nv+1Nsμ4πAiAj∫V⇀′i∫V⇀′JI^i⋅I^jRijdV′dV∂Istjtd∂t+1Ai⋅∫V′⇀i∂Φr⇀,t∂lidV=I^iAi⋅∫V′⇀iJ⇀r⇀,tσdV+∑j=1Nsμ4πAiAj∫V⇀′i∫V′⇀jI^i⋅I^jRijdV′dV∂Itjtd∂t+∑i=1Ns∑k=1Nj14πε1Sij∫Sij∫Si′ρstir⇀,tRijdS′dS.Equation (5) can be used to get the discrete vector magnetic potential and scalar potential.The terms of (6) can be regarded as a voltage drop. In equation (6) on the right, there are three terms, and they are denoted as VRi, VLi, and VC:(8)VRi=I^iAi⋅∫V′⇀iE⇀r⇀,tdV=liσAiIit,VLi=∑j=1Nsμ4πAiAj∫V⇀′i∫V′⇀jI^i⋅I^jRdV′dV∂Itjtd∂t,VC=ϕijt=∑i=1Ns∑k=1Nj14πε1SijQiSi∫Sij∫Si′1RijdS′dS.According to the definition equationsR=V/I, VL=Ldi/dt, and VC=Q/C, we get the following.Partial resistance:(9)Rpi=liσAi.Partial inductance:(10)Lpi,j=μ04πAiAj∫V⇀′i∫V′⇀jI^i⋅I^jRdV′dV,where i=j indicates partial self-inductance and i≠j indicates partial mutual inductance between cells i and j.Partial capacitance (notePpij is the potential factor):(11)Ppij=14πε1Sij1Si∫Sij∫Si11RijdS′dS,Cij=Ppij−1,V0=VRi+VLi+VCi,=RpIi+LpIi+ϕi−ϕj.According to equation (11), when the applied current Ii is established, the equivalent circuit corresponding to current element i is shown in Figure 1.Figure 1
PEEC model of current elementi.Considering the applied excitation source as an independent voltage sourceV0, denoted as VCi=ϕi−ϕj, (6) can be written as a KVL.In the PEEC modeling, the current is satisfied:(12)Ii=IL+IC.Equation (11) can get expressions in frequency domain as(13)Iiω=jωP−1ωΦω+ILω,V0ω=RpIiω+jωLpIiω+ϕiω−ϕjω.The system of frequency-domain solution matrix equations of the equivalent circuit is obtained by combining (11) and (12).(14)sPd−1I^+MALTAL−R+sLΦnIL=I^+MIiVi,where s=jω is the angular frequency operator, AL is the adjacency matrix, and the internal values are(15)aij=1,currentIiflows out of nodej,−1,currentIiflows into nodej,0,others,where R is the partial resistance matrix and L is the partial inductance matrix.(16)R=R110⋯00R22⋯0⋮⋮⋯⋮00⋯RNN,L=L11L12⋯L1NL21L22⋯L2N⋮⋮⋯⋮LN1LN2⋯LNN,where Pp is the potential coefficient matrix, which is decomposed into the diagonal Pd and nondiagonal array Pp_.(17)Pp=Pp11Pp12⋯Pp1NsPp21Pp22⋯Pp2Ns⋮⋮⋯⋮PpNs1PpNs2⋯PpNsNs,Pd=Pp110⋯00Pp22⋯0⋮⋮⋯⋮00⋯PpNsNs,Pp_=0Pp12⋯Pp1NsPp210⋯Pp2Ns⋮⋮⋯⋮PpNs1PpNs2⋯0,where I^ is a unit matrix of the same order as the potential coefficient matrix.(18)I^+M=1Pp12Pp11⋯Pp1NsPp11Pp22Pp221⋯Pp2NsPp22⋮⋮⋯⋮PpNs1PpNsNsPpNs2PpNsNs⋯1,where Φn is the nodal potential matrix, IL is the inductor current matrix, Ii is the input current source column vector matrix, and Vi is the input voltage source column vector matrix.(19)Φn=φ1φ2⋯φnT,IL=IL1IL2⋯ILNT,Ii=I1I2⋯InT,Vi=V1V2⋯VNT.


## 3. Establishment of PEEC Model for Steel Rails

### 3.1. 3D Physical Model of Steel Rails

Figure2 depicts a three-dimensional physical model of the rail self-impedance calculation. The rail is attached to the bed foundation and supports train operation while also acting as a carrier for track circuit signal transmission. Rail self-impedance includes both internal and external impedance [18]. Internal impedance is only affected by the rail’s material, current amplitude, and frequency. Longitudinal impedance (the rail and the earth are ideal conductors) and impedance between the rail and the earth circuit are the two types of external impedance for the rail (earth is a lossy conductor). The electrical environment affects the exterior impedance. Because the characteristics and electrical environment of two parallel rails are usually the same, only one rail is considered when calculating self-impedance.Figure 2
Simplified model of single-line track.


### 3.2. PEEC Model of Steel Rails

Different types of rails have different materials and appearance dimensions, and this paper selects P60-type rails as an example to calculate the self-impedance of rails by the PEEC method. The rail cross section is irregular. Orthogonal hexahedral partitioning and rectangular hexahedral partitioning need to be equivalent to the rail shape. Therefore, under the premise of ensuring accuracy, the cross section will be equivalent according to the rail head, rail web, and rail bottom. The rail electromagnetic parameters and equivalent dimensions are shown in Tables1 and 2, and the rail cross section and equivalent cross section are shown in Figure 3.Table 1
Material parameters of steel rails.
Material nameResistivity (Ω·m)Relative magnetic permeabilityP602.1 × 10−7100Table 2
Rail dimensions and equivalent dimensions.
ParametersRail typeP60Cross-sectional area S/cm277.45Cross-sectional circumference C/cm67.00Equivalent effectRail headH h/mm40.50W w/mm73.00Rail bellyH h/mm63.00W w/mm16.50Rail bottomH h/mm25.00W w/mm150.00Figure 3
P60 rail cross section and equivalent section diagram. (a) P60 rail cross section. (b) Equivalent section diagram.
(a)(b)According to the principle of orthogonal hexahedral and rectangular surface profile, the equivalent rail is profiled, and since the length of the rail is much larger than its cross-sectional circumference, its longitudinal profile is focused on. The conductive metal material in high frequency will produce skin effect, which result the distribution of current in the rail cross-sectional is not uniform, so the calculation of single rail self-impedance should consider the effect of skin effect. When the transverse of rail is dissected, mesh size should be less than the rail skin depth determined by frequency. According to reference [25], the skin depth calculation formula can be obtained as(20)δ=1πfμσ.Now assume that the equivalent rail is divided intoN parts along the longitudinal section and M parts along the cross section, and the corresponding PEEC model is shown in Figure 4.Figure 4
PEEC model considering skinning effect (not decoupled).Assuming that the dissected cell isΔx long, Δy wide, and Δz high along the current direction, as shown in Figure 5. The effect on the skinning effect can be regarded as an equivalent impedance Zs with the skinning depth.(21)Zs=Rs+jωLs,Rs=ΔxσδΔy,Ls=ΔxωσδΔy.Figure 5
Schematic diagram of cell skinning effect.Two adjacent cells build decoupled PEEC models as shown in Figure6.Figure 6
Decoupling PEEC of two adjacent metacells.The partial resistance, partial inductance, and partial capacitance at each node in the decoupled PEEC model areR, L, and C, respectively (capacitance to ground). The controlled voltage source is caused by the partial mutual inductance generated by the coupling of different inductive units, and the controlled current source is caused by the partial mutual capacitance between different capacitive units.


## 3.1. 3D Physical Model of Steel Rails

Figure2 depicts a three-dimensional physical model of the rail self-impedance calculation. The rail is attached to the bed foundation and supports train operation while also acting as a carrier for track circuit signal transmission. Rail self-impedance includes both internal and external impedance [18]. Internal impedance is only affected by the rail’s material, current amplitude, and frequency. Longitudinal impedance (the rail and the earth are ideal conductors) and impedance between the rail and the earth circuit are the two types of external impedance for the rail (earth is a lossy conductor). The electrical environment affects the exterior impedance. Because the characteristics and electrical environment of two parallel rails are usually the same, only one rail is considered when calculating self-impedance.Figure 2
Simplified model of single-line track.


## 3.2. PEEC Model of Steel Rails

Different types of rails have different materials and appearance dimensions, and this paper selects P60-type rails as an example to calculate the self-impedance of rails by the PEEC method. The rail cross section is irregular. Orthogonal hexahedral partitioning and rectangular hexahedral partitioning need to be equivalent to the rail shape. Therefore, under the premise of ensuring accuracy, the cross section will be equivalent according to the rail head, rail web, and rail bottom. The rail electromagnetic parameters and equivalent dimensions are shown in Tables1 and 2, and the rail cross section and equivalent cross section are shown in Figure 3.Table 1
Material parameters of steel rails.
Material nameResistivity (Ω·m)Relative magnetic permeabilityP602.1 × 10−7100Table 2
Rail dimensions and equivalent dimensions.
ParametersRail typeP60Cross-sectional area S/cm277.45Cross-sectional circumference C/cm67.00Equivalent effectRail headH h/mm40.50W w/mm73.00Rail bellyH h/mm63.00W w/mm16.50Rail bottomH h/mm25.00W w/mm150.00Figure 3
P60 rail cross section and equivalent section diagram. (a) P60 rail cross section. (b) Equivalent section diagram.
(a)(b)According to the principle of orthogonal hexahedral and rectangular surface profile, the equivalent rail is profiled, and since the length of the rail is much larger than its cross-sectional circumference, its longitudinal profile is focused on. The conductive metal material in high frequency will produce skin effect, which result the distribution of current in the rail cross-sectional is not uniform, so the calculation of single rail self-impedance should consider the effect of skin effect. When the transverse of rail is dissected, mesh size should be less than the rail skin depth determined by frequency. According to reference [25], the skin depth calculation formula can be obtained as(20)δ=1πfμσ.Now assume that the equivalent rail is divided intoN parts along the longitudinal section and M parts along the cross section, and the corresponding PEEC model is shown in Figure 4.Figure 4
PEEC model considering skinning effect (not decoupled).Assuming that the dissected cell isΔx long, Δy wide, and Δz high along the current direction, as shown in Figure 5. The effect on the skinning effect can be regarded as an equivalent impedance Zs with the skinning depth.(21)Zs=Rs+jωLs,Rs=ΔxσδΔy,Ls=ΔxωσδΔy.Figure 5
Schematic diagram of cell skinning effect.Two adjacent cells build decoupled PEEC models as shown in Figure6.Figure 6
Decoupling PEEC of two adjacent metacells.The partial resistance, partial inductance, and partial capacitance at each node in the decoupled PEEC model areR, L, and C, respectively (capacitance to ground). The controlled voltage source is caused by the partial mutual inductance generated by the coupling of different inductive units, and the controlled current source is caused by the partial mutual capacitance between different capacitive units.


## 4. Algorithm Feasibility Verification

Rail not only is the transmission carrier of the signal current but also is the carrier of the traction return current with hundred amperes. Because the rail material has ferromagnetic qualities, increasing current causes magnetization, which increases relative permeability. However, due to the impact of magnetic saturation, magnetic permeability will only grow to a limited level as current increases. Ampere’s loop theorem is used to compute the magnetic field strength generated by different current amplitudes at 50 Hz, and relative permeability fluctuates with current amplitude according to the functional connection between permittivity and magnetic field intensity, as shown in Table3.Table 3
Relative magnetic permeability variation law with current amplitude.
Current amplitude (A)Relative magnetic permeabilityμrCurrent amplitude (A)Relative magnetic permeabilityμrCurrent amplitude (A)Relative magnetic permeabilityμr100150400450700490200200500460800180300370600500900445The variation law of the internal impedance computed with current amplitude by the PEEC and FEM, respectively, is shown in Figures7 and 8 using a P60 rail with a length of 1 m, and the electrical and exterior dimensional data are given in Tables 1 and 2. The results are also compared with the experimental values of reference [12].Figure 7
Internal resistance varying with current amplitude.Figure 8
Internal inductance varying with current amplitude.Figures7 and 8 indicate that the PEEC technique and the FEM approach are more accurate for calculating the impedance within the rails in the region of 0 to 300 amperes, but as the current amplitude increases, the two methods begin to display a certain mistake. It can be observed that the PEEC approach can be successfully used to the calculation of rail impedance.The relative errors between the two methods and the experimental values are obtained according to (22), and the results are shown in Figures 9 and 10.(22)δ=x−x0x×100%.Figure 9
Relative error of internal resistance compared to experimental values.Figure 10
Relative error of internal inductance compared to experimental values.As shown in Figures9 and 10, the PEEC approach has a greater accuracy based on the relative error between the two methods and the experimental value. Secondly, under the same hardware conditions, the PEEC method takes 69.298071 seconds and FEM consumes 187.10479 seconds. So, PEEC has superiority in calculation accuracy and calculation efficiency.


## 5. Self-Impedance Calculation and Analysis of Steel Rails

### 5.1. Self-Impedance Calculation of Steel Rails

Rail impedance includes the internal impedance and external impedance [18]. The PEEC method do not suit to calculate the external impedance due to the infinite earth cannot be equivalent by this method. As a result, the Carson formula may be used to determine the rail’s exterior impedance under the effect of the earth [4]. Figure 11 shows overhead line at ha above the earth.Figure 11
Single overhead line high above groundha.According to Carson’s formula, the impedance of the overhead line in Figure11 can be expressed as(23)Zself=Zinternal+Zpcg+Zcor,where the last two terms Zpcg+Zcor in the (23) represent the external impedance, representing the longitudinal impedance and the ground loop impedance, respectively.(24)Zaapcg=jωμ02πln2hara,(25)Zaacor=jωμ0π∫0∞e−2haλt2+jγ2+tdt,where ω=2πf is the angular frequency (f is current frequency); ha is the height of the overhead line above the ground; ra is the radius of the overhead line a; t is the integration variable; and γ is the magnitude of the complex propagation constant.The complex propagation constantγ¯ and its magnitude γ are(26)γ¯=jωμ0σd,γ=ωμ0σd,where σd is the earth conductivity.From (25), it can be seen that the overhead line-ground loop impedance expression contains an infinite upper limit of the oscillation integral, and the segmented linear approximation method is used to solve it [7].The self-impedance of unit length P60 rail is calculated by the above method, and results are shown in Figure12. It is noted that all the electrical parameters and external dimension data are from Tables 1 and 2. The rail along the longitudinal length is divided into 100 parts, and that along the cross section is divided into 25 parts. For each part of the cell cross-sectional area of 3.098×10−4 m2, the frequency at 10000 Hz skinning depth is 2.3064×10−4 m based on (20).Figure 12
Variation of self-impedance of rails with frequency.Figure12 shows that when the frequency is 10−37Hz close to 0Hz, the track self-impedance is 2.711×10−4Ω, which is nearly equivalent to the DC resistance estimated using the PEEC approach. It again verifies the correctness of the PEEC method for solving the rail self-impedance.


### 5.2. Analysis of Influencing Factors

#### 5.2.1. The Effect of Frequency on the Self-Impedance of Rails

According to Figure12, it can be seen that the modulus value of rail self-impedance increases with frequency from 0 to 10000 Hz. Due to the ferromagnetic properties of the rail, the current is affected by the skin effect in high frequencies, which causes the resistance to increase with frequency and the inductance to decrease with frequency.


#### 5.2.2. The Effect of Rail Height above Ground and Earth Conductivity on the Self-Impedance of Rails

Figures13 and 14 show the result of self-impedance when the highness of the rail above the ground is in the range of 0m≤h≤1m and the earth conductivity is in the range of 10−3S/m≤σ≤1S/m.Figure 13
The effect of rail height from the ground on the impedance of rails.Figure 14
The effect of earth conductivity on the impedance of steel rails.The results in Figure13 demonstrate that when the frequency is 50 Hz or 500 Hz, the rail self-impedance nearly does not change when the rail’s height from the ground increases. When the frequency is between 5000 and 10000 Hz, the rail self-impedance decreases as the height of the rail from the ground rises. The reason is that energy is lost in the rail-ground loop, which result in higher rail impedance when the rail is closer the earths.Figure14 shows that the rail self-impedance modulus value decreases significantly with the increase of earth conductivity, which indicates that the influence of earth conductivity on rail self-impedance is larger. At the same time, it shows that the influence of the highness of the rail above the ground on the self-impedance also increases gradually when the earth’s conductivity is larger.


## 5.1. Self-Impedance Calculation of Steel Rails

Rail impedance includes the internal impedance and external impedance [18]. The PEEC method do not suit to calculate the external impedance due to the infinite earth cannot be equivalent by this method. As a result, the Carson formula may be used to determine the rail’s exterior impedance under the effect of the earth [4]. Figure 11 shows overhead line at ha above the earth.Figure 11
Single overhead line high above groundha.According to Carson’s formula, the impedance of the overhead line in Figure11 can be expressed as(23)Zself=Zinternal+Zpcg+Zcor,where the last two terms Zpcg+Zcor in the (23) represent the external impedance, representing the longitudinal impedance and the ground loop impedance, respectively.(24)Zaapcg=jωμ02πln2hara,(25)Zaacor=jωμ0π∫0∞e−2haλt2+jγ2+tdt,where ω=2πf is the angular frequency (f is current frequency); ha is the height of the overhead line above the ground; ra is the radius of the overhead line a; t is the integration variable; and γ is the magnitude of the complex propagation constant.The complex propagation constantγ¯ and its magnitude γ are(26)γ¯=jωμ0σd,γ=ωμ0σd,where σd is the earth conductivity.From (25), it can be seen that the overhead line-ground loop impedance expression contains an infinite upper limit of the oscillation integral, and the segmented linear approximation method is used to solve it [7].The self-impedance of unit length P60 rail is calculated by the above method, and results are shown in Figure12. It is noted that all the electrical parameters and external dimension data are from Tables 1 and 2. The rail along the longitudinal length is divided into 100 parts, and that along the cross section is divided into 25 parts. For each part of the cell cross-sectional area of 3.098×10−4 m2, the frequency at 10000 Hz skinning depth is 2.3064×10−4 m based on (20).Figure 12
Variation of self-impedance of rails with frequency.Figure12 shows that when the frequency is 10−37Hz close to 0Hz, the track self-impedance is 2.711×10−4Ω, which is nearly equivalent to the DC resistance estimated using the PEEC approach. It again verifies the correctness of the PEEC method for solving the rail self-impedance.


## 5.2. Analysis of Influencing Factors

### 5.2.1. The Effect of Frequency on the Self-Impedance of Rails

According to Figure12, it can be seen that the modulus value of rail self-impedance increases with frequency from 0 to 10000 Hz. Due to the ferromagnetic properties of the rail, the current is affected by the skin effect in high frequencies, which causes the resistance to increase with frequency and the inductance to decrease with frequency.


### 5.2.2. The Effect of Rail Height above Ground and Earth Conductivity on the Self-Impedance of Rails

Figures13 and 14 show the result of self-impedance when the highness of the rail above the ground is in the range of 0m≤h≤1m and the earth conductivity is in the range of 10−3S/m≤σ≤1S/m.Figure 13
The effect of rail height from the ground on the impedance of rails.Figure 14
The effect of earth conductivity on the impedance of steel rails.The results in Figure13 demonstrate that when the frequency is 50 Hz or 500 Hz, the rail self-impedance nearly does not change when the rail’s height from the ground increases. When the frequency is between 5000 and 10000 Hz, the rail self-impedance decreases as the height of the rail from the ground rises. The reason is that energy is lost in the rail-ground loop, which result in higher rail impedance when the rail is closer the earths.Figure14 shows that the rail self-impedance modulus value decreases significantly with the increase of earth conductivity, which indicates that the influence of earth conductivity on rail self-impedance is larger. At the same time, it shows that the influence of the highness of the rail above the ground on the self-impedance also increases gradually when the earth’s conductivity is larger.


## 5.2.1. The Effect of Frequency on the Self-Impedance of Rails

According to Figure12, it can be seen that the modulus value of rail self-impedance increases with frequency from 0 to 10000 Hz. Due to the ferromagnetic properties of the rail, the current is affected by the skin effect in high frequencies, which causes the resistance to increase with frequency and the inductance to decrease with frequency.


## 5.2.2. The Effect of Rail Height above Ground and Earth Conductivity on the Self-Impedance of Rails

Figures13 and 14 show the result of self-impedance when the highness of the rail above the ground is in the range of 0m≤h≤1m and the earth conductivity is in the range of 10−3S/m≤σ≤1S/m.Figure 13
The effect of rail height from the ground on the impedance of rails.Figure 14
The effect of earth conductivity on the impedance of steel rails.The results in Figure13 demonstrate that when the frequency is 50 Hz or 500 Hz, the rail self-impedance nearly does not change when the rail’s height from the ground increases. When the frequency is between 5000 and 10000 Hz, the rail self-impedance decreases as the height of the rail from the ground rises. The reason is that energy is lost in the rail-ground loop, which result in higher rail impedance when the rail is closer the earths.Figure14 shows that the rail self-impedance modulus value decreases significantly with the increase of earth conductivity, which indicates that the influence of earth conductivity on rail self-impedance is larger. At the same time, it shows that the influence of the highness of the rail above the ground on the self-impedance also increases gradually when the earth’s conductivity is larger.


## 6. Conclusion

Considering the effect of skin effect, the PEEC method is used to establish the model of P60 rail, and the equations of the PEEC model are solved. Carson’s theoretical formula and PEEC are combined. The self-impedance of the rail is determined. Based on the above research, the following conclusions are obtained:(1)
The PEEC method was used to calculate the internal impedance of a unit length of P60 rail. Compared with the experimental values, the relative errors of the internal resistance and internal inductance were less than 4% in the frequency of 0∼10000 Hz, which prove that the PEEC method can be well applied to the calculation of rail self-impedance; it also includes a theoretical calculation approach for calculating rail self-impedance.(2)
When experimental data are compared to the findings of the PEEC approach and the FEM for determining rail impedance, the PEEC method has greater accuracy and takes less time to compute.(3)
In 0∼10000 Hz, rail self-impedance rises with frequency. The rail’s self-impedance decreases as the rail height increases in high frequency. With the rise in earth conductivity, the rail self-impedance decreases. When the earth’s conductivity is greater, the influence of rail highness above the ground on rail impedance is correspondingly greater.



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*Source: 2786881-2022-05-04.xml*