## 5 Conclusions  
This paper is concerned with the AT system in HSR and proposes a spatial magnetic field description method, combined with the ideas of a multi-conductor loop circuit and iteration. This method cannot only directly reflect the complete current distribution of a traction network, but also deduce the exact traction network integrated impedance.  
First, due to the limitations of the existing approach, the hypothesis that the earth is the reference conductor of a structuring loop circuit matrix is cancelled. To satisfy the condition of complete and accurate spatial magnetic field description, all the real traction network loop circuits and their self and mutual inductance influences are considered in order to list the loop circuit matrix of the AT system.  
Next, considering the actual electrical structure of the AT system, different description processes are divided, respectively, for the right side of the load (Space 1) and the left side of the load (Space 2). Considering that the two types of return conductors have different potential differences with the transmission conductor in the AT system, certain current ratios deduced in existing studies are introduced. For Space 1, after directly introducing the current ratio into the loop circuit matrix and performing some matrix transformations on this basis, the complete current distribution and the impedance of the traction network can be obtained. For Space 2, after directly introducing the current ratio into the loop circuit matrix and performing some matrix transformations on this basis, the deduced results are actually inaccurate, for the reason that the introduced current ratio of Space 2 is approximate. Therefore, an iterative scheme is adopted for the deduction of Space 2. According to the transformer ratio of the AT substation, a constraint condition of circuit voltages is determined. By approximately searching and adjusting the current ratio to satisfy the constraint condition of circuit voltages, accurate current distribution and traction network integrated impedance are finally obtained.  
Finally, experimental verification was undertaken. The results demonstrate that the presented method has the capability to depict the exact current distribution and traction network integrated impedance of an AT system.  
Further investigations on the improvement of the spatial magnetic field description method of an AT system are necessary. For example, it is feasible to obtain the exact rail potential distribution if the leakage conductance between rail and earth is considered in the establishment of the loop circuit matrix. Also, some engineering applications of the description method, such as testing software development, can be further studied.  
Conceptualization, Y.D. and K.H.; Methodology, Y.D.; Software, D.S.; Validation, Y.D., K.H. and Z.L.; Formal Analysis, Y.D.; Investigation, Y.D.; Resources, Z.L.; Data Curation, D.S.; Writing-Original Draft Preparation, K.H.; Writing-Review & Editing, Z.L.; Visualization, Y.D.; Supervision, Z.L.; Project Administration, Y.D.; Funding Acquisition, Y.D.  
This research was funded by National Nature Science Foundation of China [U1434203, U1734202], Sichuan Province Youth Science and Technology Innovation Team [2016 TD0012] and Research Project of China Railway Eryuan Engineering Group Co. Ltd. [KYY2017025(17-18)]  
This work is supported by the National Nature Science Foundation of China (U1434203, U1734202), Sichuan Province Youth Science and Technology Innovation Team (2016 TD0012), and Research Project of China Railway Eryuan Engineering Group Co. Ltd. (KYY2017025(17-18)).  
The authors declare no conflict of interest.