## 1 Introduction  
China's high-speed trains (HST) has been developed rapidly in recent years. As an important system of a high-speed railway (HSR), the traction network directly influences the reliable power supply of trains. Focusing on the traction network, the short-circuit influence and the voltage loss are two important parts of the power-supply capability. In the actual operation of an HSR, the short-circuit catenary fault is easily caused by an accidental collision of a bird with the conductor, an internal fault of the train, etc. The voltage loss of the traction network is heavily influenced by the power supply mode of the traction network. Therefore, it is necessary to effectively assess the power supply capability of the traction network on the basis of establishing an accurate traction network mathematical model. The power supply capability is reflected by the traction network integrated impedance. Performing complete and accurate spatial magnetic field description is a premise for obtaining exact traction network integrated impedance. Aiming at the traction network, many scholars have tried to establish its mathematical model by applying multi-conductor transmission-line (MTL) theory. Existing models mainly include a unified chain circuit model applicable to different power supply modes [1; 2], harmonic models [3; 4], MTL models of various traction networks, e.g., an auto-transformer (AT) supply system and a novel cable supply system [5; 6; 7]. Some application studies have been also done. In [5], the magnetic fields and harmonics in the surroundings of the AT traction network were analyzed. In [8; 9], the electromagnetic compatibility of the traction network was assessed. In [10], a strategy for suppressing low-frequency oscillation has been proposed and its effectiveness was proved by a model simulating a HST-traction network electrical coupling system. Aimed at the AT system with protection wire, a merge process of protection wire and rail was performed and the rail potential was analyzed with a decoupling method in [11]. Moreover, many researchers have studied the electromagnetic interference caused by a common lightning strike [12], pantograph detachment arc [13], etc. Meanwhile, with the help of theoretical analyses, software simulations or experimental measurements, models of the arc, HST, etc., are established and introduced into various traction network mathematical models to analyse accurately interference sources such as arcing phenomenon, vehicle body over-voltages, pantograph over-voltages [14; 15]. Focusing on railway lines on special roads such as stations, tunnels and viaducts, some electromagnetic transient models were also studied [16; 17; 18; 19; 20]. In [18], the traction network system of the viaduct path is equivalent to the lumped parameter model and the viaduct's electrical coupling is considered in modelling. In [19; 20], the influences of the arc-burning phenomenon of insulated rail joints were investigated by state-space model analyses and chain modelling simulations, respectively. Combined with MTL theory, the inductive and capacitive couplings of conductors in the station path are reflected in the chain model of the traction network. Nevertheless, in the aspect of spatial magnetic field description, the application study of traction network models has been insufficient [21]. According to basic electromagnetic theory, obtaining complete and accurate current distribution is the premise for realizing complete and accurate spatial magnetic field description. For the AT system that is widely applied to HSR, some scholars have investigated the current distributions by equaling the AT system to a simple topology [22; 23;  
In addition, since the currents of Loop circuits 3-4 are not generated by the excitation source, the deduced interlinkage flux of Loop circuits 3-4 and Loop circuits 1-2 is different from the inter linkage flux of the real loop circuits. This leads to the imprecise description of the spatial magnetic field.