## 1 Introduction
Thus, the deductions of current distribution and integrated impedance (marked as \(Z_{0}\) here) implement the process of description. As for \(Z_{0}\), this is the integrated impedance of all the traction network loop circuits in parallel from the location of the traction substation to the location of the load or short-circuit point. As the load is located at the path between the TSS and ATS in Figure 3a, unit length \(Z_{0}\) is obtained by:  
\[Z_{0}\ =\ \frac{u_{T}+u_{F}}{(i_{T}+i_{F})x}\ =\ \frac{u_{T}+u_{F}}{i^{\prime}x} \tag{1}\]  
As the load is located at the path between adjacent ATSs in Figure 3b, unit length \(Z_{0}\) is obtained by:  
\[Z_{0}\ =\ \frac{u_{T}+u_{F}}{(i_{T}+i_{F})(D+x)}\ =\ \frac{u_{T}+u_{F}}{i^{ \prime}(D+x)} \tag{2}\]  
In addition, Equation (3) shows the relation among \(u_{T}\), \(u_{F}\), \(i^{\prime}\), \(i_{T}\), \(i_{G}\), \(i_{F}\)[22]. \[\left\{ \begin{array}{l}u_{T}\ =\ u_{F}\\ i^{\prime}\ =\ i_{T}+i_{F}\\ i_{T}\ =\ i_{G}+i_{F}\end{array} \right. \tag{3}\]  
In the segment loop circuit, return currents are borne by F and N. The relation among \(i_{R1}\), \(i_{R2}\) and \(i_{F1}\) is often expressed as Equation (4) according to [22]. Figure 3: Currents of AT double-track system in a high-speed railway (HSR). \[\left\{ \begin{array}{l}i_{R1}\ =\ 2\left(\frac{D}{x}-1\right)\cdot i_{F1}\\ i_{R2}\ =\ 2i_{F1}\end{array} \right. \tag{4}\]  
According to Equation (4), when the load is running in the segment loop circuit, the distribution proportions of return currents in N and F will constantly change. At the beginning position of the segment loop circuit (i.e., \(x=0\) in Equation (4)), the return current is fully borne by N. At the terminal position of the segment loop circuit (i.e., \(x=D\) in Equation (4)), the return current is fully borne by F. The current distribution and the segment loop circuit impedance are equal to that of the long loop circuit. So, to get the integrated impedance and current distribution of the whole AT double-track system, it is only necessary to obtain the deduction for the segment loop circuit. In addition, in terms of the Space 1 and the Space 2 seen in Figure 3, the deductions are separately made at first. Since the relation of \(i_{R1}\), \(i_{F1}\) and the relation of \(i_{R2}\), \(i_{F1}\) are different, the deduction processes of Space 1 and Space 2 are illustrated, as follows, separately. Deduction Scheme Aimed at the Condition that the Load Exists in Up Traction Network or Down Traction Network  
Because the parameters and the structure of the up and down traction networks are the same, the solution processes and results of current distribution and traction network integrated impedance are entirely identical. Thus, only the situation that the load exists in the up traction network is considered. When the load is in the up traction network, the loop circuits of 1-20 are considered, as listed in Table 1. According to the relation between unit-length voltage drop and unit-length flux linkage of each loop circuit, when the source is a sinusoidal excitation, this can be obtained below. \[\left[ \begin{array}{c}\Delta u_{1}\\ \Delta u_{2}\\ \vdots\\ \Delta u_{i}\\ \vdots\\ \Delta u_{20}\end{array} \right]\ =\ j\omega\left[ \begin{array}{cccccc}l_{1-1}&l_{1-2}& \cdots&l_{1-j}&\cdots&l_{1-20}\\ l_{2-1}&l_{2-2}&\cdots&l_{2-j}&\cdots&l_{2-20}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ l_{i-1}&l_{i-2}&\cdots&l_{i-j}&\cdots&l_{i-20}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ l_{20-1}&l_{20-2}&\cdots&l_{20-j}&\cdots&l_{20-20}\end{array} \right]\left[ \begin{array}{c}i_{1}\\ i_{2}\\ \vdots\\ i_{j}\\ \vdots\\ i_{20}\end{array} \right] \tag{5}\]  
where \(\omega\) is angular velocity, \(\Delta u_{i}\) is voltage drop of loop circuit \(i\), \(i_{j}\) is current of loop circuit \(j\), \(l_{i\text{-}j}\) (\(i=j\)) is self-inductance of each loop circuit, and \(l_{i\text{-}j}\) (\(i\
eq j\)) is mutual inductance between loop circuit \(i\) and loop circuit \(j\) (\(i,j=1,2\ldots\) 20).