## 1 Introduction
Analysis of Auto-Transformer (AT) Double Track System Based on Multi-Conductor Transmission-Line (MTL) Theory  
The cross section of standard 2 \(\times\) 25-kV AT railway system line is depicted in Figure 2, where the contact wire, carrier cable, two rails, protective wire, integrated earth wire and positive feeder in up traction network (down traction network) are marked as JW1(JW2), CW1(CW2), R1(R3), R2(R4), PW1 (PW2), EW1(EW2) and FW1(FW2), respectively; and the earth is marked as E. Because each loop circuit is constituted by a transmission conductor and a return conductor while all the other conductors and the earth are return conductors, the standard railway system line in Figure 2 has 40 loop circuits. The numbers of all the loop circuits are shown in Table 1.  
In the AT system, return conductors can be divided into magnetic field conversion return conductors and electric field return conductors. FW1, FW2 in Figure 2 belong to the former because they constitute loop circuits via the magnetic field conversion of the auto-transformer substation (ATS). R1, R2, PW1, EW1, R3, R4, PW2, EW2, E belong to the latter because they constitute loop circuits by connecting with each other through conductors/leakage conductance, or contacting the rail with the HST's wheel. The AT system is composed of two types of AT segments. One is the path between the traction substation (TSS) and the ATS. The other is the path between the adjacent ATSs. For the two types of AT segments, the path at which load (expressed as HST here) is located is called the segment loop \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Conductors** & **R1** & **R2** & **PW1** & **EW1** & **R3** & **R4** & **PW2** & **EW2** & **E** & **FW1** & **FW2** \\ \hline JW1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & - \\ CW1 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & - \\ JW2 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & - & 30 \\ CW2 & 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & - & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Loop circuit number of conductors in standard 2 \(\times\) 25 kV auto-transformer (AT) railway system line. Figure 2: Typical cross section of auto-transformer (AT) double-track system. circuit while the path at which the load is not located is called the long loop circuit. That is, when the load is located at the path between the TSS and the ATS, the path between the TSS and the ATS is the segment loop circuit while the other path is the long loop circuit. When the load is located at the path between the ATSs, the path between the TSS and the ATS is the long loop circuit while the other path is the segment loop circuit. Figure 3 depicts the currents of the AT system when assuming that the transmission conductors, electric field return conductors and magnetic field conversion return conductor are marked as T, N and F. In Figure 3, \(i_{T1}\) and \(i_{T2}\) are currents of T at the left and right sides of the load, respectively; \(i_{R1}\) and \(i_{R2}\) are currents of N at the left side and right side of load; \(i_{F1}\) is current of F at path of segment loop circuit; \(i^{\prime}\) is load current; \(i_{T}\), \(i_{F}\) and \(i_{G}\) are respectively total currents of T, F and N in the up and down traction network; \(u_{T}\) is potential to earth of positive bus-bar in the TSS; \(u_{F}\) is potential to earth of the negative bus-bar in the TSS. Both the distance between the TSS and ATS and the distance between adjacent ATSs are assumed to be \(D\). When the load is located at the path between the TSS and ATS, \(x\) is the distance between the load and TSS at the left side; when the load is located at the path between adjacent ATSs, \(x\) is the distance between the load and ATS at the left side. For the complete and accurate spatial magnetic field description of the AT system, obtaining complete and exact current distribution is the premise, and solving the precise traction network integrated impedance is an important aim.