## 1 Introduction
Furthermore, it can be demonstrated from Figure 1 that the currents of Loop circuits 1-4 are namely the currents of the carrier cable, contact cable, catenary, rail and return cable. However, the currents of Loop circuits 1-4 are only the partial currents of the carrier cable, contact cable, catenary, rail and return cable in actuality. Taking the carrier cable current an example, this includes not only the current of Loop circuit 1, but also the currents of the loop circuit that is constituted by carrier cable and rail and the loop circuit that is constituted by the carrier cable and return cable. So, the conventional approach lacks the complete description. Finally, according to the conventional approach, the sum of currents of Loop circuits 3-4 is assumed to be equal to that of Loop circuits 1-2 in Figure 1. Obviously, the earth's own current is ignored in this relation. The actual relation should be that the sum of the currents of Loop circuits 3-4 and the self-current of the earth is equal to that of Loop circuits 1-2. Neglecting the earth self-current also illustrates the incompleteness of the conventional approach. For the reason that the real loop circuit is composed of the transmission conductor and the return conductor, the total number of real loop circuits in a simple direct power supply system is six, including Loop circuit 1\({}^{\prime}\), Loop circuit 2\({}^{\prime}\)... Loop circuit 6\({}^{\prime}\) of Figure 1. If all the real loop circuits are considered in the calculation of traction network current distribution, it is possible to gain a complete and accurate spatial magnetic field description. In addition, as for the AT system, two types of return conductors exist in it. The electrical structure of the AT system is relatively more complex than that of a simple direct power supply system. Considering that there is also no related research that considers the detailed electrical structure of the AT system so far, it is necessary to have a complete and accurate spatial magnetic field description of AT system, after considering all the real traction network loop circuits and the power supply characteristics of the AT system. In this paper, a spatial magnetic field description method of the AT system is presented. It not only adopts the traction network mathematical model based on MTL theory, but also combines the actuality of the detailed electrical structure of AT system. Starting from fundamental considerations in theory, the traditional assumption that the earth is the reference conductor of all the conductors is canceled and all the real traction network loop circuits are considered as fundamental spatial magnetic field  
Figure 1: Diagram of the formed loop circuits when assuming the earth as the reference conductor and all the real loop circuits for a simple direct power supply system. units. Considering that two types of return conductors exist and they have different voltages to the transmission conductor in the AT system, a fundamental relation formula of return conductor current and transmission conductor current is introduced. On this basis, an iterative calculation approach is adopted to realize the exact spatial magnetic field unit. The rest of this paper is organized as follows. Section 2 shows the analysis of an AT double-track system based on MTL theory. Then, the common situation that the load is running in the up traction network or the down traction network is taken as an example to illustrate the description processes of the proposed description method in Section 3. In Section 4, the effectiveness and accuracy of the proposed method are proved by real-life experimental verification. Section 5 draws some conclusions.