Spatial Magnetic-Field Description Method Aimed at 2 × 25 kV Auto-Transformer Power Supply System in High-Speed Railway

: Complete and accurate spatial magnetic ﬁeld description is the premise of effectively assessing the power supply capability of a high-speed railway (HSR). Its evaluation indicators are the current distributions and the integrated impedance of traction network. This paper proposes a spatial magnetic-ﬁeld description method for the auto-transformer (AT) power supply system. Due to the limitations of previous approaches, all the real loop circuits of the AT system are considered for structuring a loop circuit matrix. At ﬁrst, different description processes are divided, respectively, into those for the right side and the left side of the load. Then, considering that two types of return conductors exist in an AT system, a certain current ratio deduced in existing studies is introduced. As the introduced current ratio of the left side of the load is approximate, an iterative scheme is adopted. By constantly adjusting current ratio to satisfy a constraint condition of circuit voltages, accurate traction network impedance is obtained. Last, in order to verify the effectiveness of the proposed description method, two real-life experiments in a Chinese HSR line are performed, which indicate that the proposed method can not only directly reﬂect complete and accurate current distribution, but also deduce the exact traction network integrated


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 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.
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′, Loop circuit 2′… Loop circuit 6′ 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 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.
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 , Loop circuit 2 . . . Loop circuit 6 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 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 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.

Analysis of Auto-Transformer (AT) Double Track System Based on Multi-Conductor Transmission-Line (MTL) Theory
The cross section of standard 2 × 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  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 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 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. 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 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, iT1 and iT2 are currents of T at the left and right sides of the load, respectively; iR1 and iR2 are currents of N at the left side and right side of load; iF1 is current of F at path of segment loop circuit; i′ is load current; iT, iF and iG are respectively total currents of T, F and N in the up and down traction network; uT is potential to earth of positive bus-bar in the TSS; uF 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. Thus, the deductions of current distribution and integrated impedance (marked as Z0 here) implement the process of description. As for Z0, 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 Z0 is obtained by: ( ) As the load is located at the path between adjacent ATSs in Figure 3b, unit length Z0 is obtained by: In addition, Equation (3) shows the relation among uT, uF, i′, iT, iG, iF [22]. 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. 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: As the load is located at the path between adjacent ATSs in Figure 3b, unit length Z 0 is obtained by: In addition, Equation (3) shows the relation among 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].
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.
where ω is angular velocity, ∆u i is voltage drop of loop circuit i, i j is current of loop circuit j, l i-j (i = j) is self-inductance of each loop circuit, and l i-j (i = j) is mutual inductance between loop circuit i and loop circuit j (i, j = 1, 2 . . . 20).

Deduction for Space 1
The i R2 = 2i F1 in Equation (4), i.e., the total currents of N are two times that of F. According to Table 1, considering that the total current of R1, R2, R3, R4, PW1, EW1, PW2, EW2, E is i R2 and the current of FW1 is i F1 , it is reasonable that In the synthesis of above relations, Equation (6) can be gained.
Meanwhile, owing to the fact that the unit-length voltage drop between JW1 and FW1 is equal to the unit-length voltage drop between CW1 and FW1, this can be formulated as: Then, by introducing Equations (6) and (7), the subtraction of ∆u 10 , ∆u 20 is performed and the current ratio of N and F is introduced in Equation (5). Equation (5) is transformed into Equation (8), where a i = l 10-i − l 20-i (i = 1, 2, · · · , 20) in the tenth row of inductance matrix.
Assuming that the total current of transmission conductors or return conductors is i 0 and the current distribution coefficient of each loop circuit of Space 1 is .. ∆u 9 , ∆u 11 , ∆u 12 ... ∆u 19 are the unit-length potential difference between T and N and they can be seen as equal to each other. In view of this, ∆u 1 = . . . = ∆u 9 = ∆u 11 = . . . = ∆u 19 = c 1 (c 1 is arbitrary constant) is set. Then, focusing on Equation (8), by performing an inverse matrix transformation and introducing i i = k i i 0 and ∆u 1 = . . . = ∆u 9 = ∆u 11 = . . . = ∆u 19 = c 1 , Equation (8) is transformed into Equation (9).
The traction network loop circuits can be divided into the overhead loop circuit and earth return loop circuit. The former is composed only of conductors while the latter is composed of the conductor and the earth. For the two types of loop circuits, the calculation formulas of the inductance matrix coefficients in Equation (5) are different. As for the overhead loop circuit, the inductance coefficients are calculated by using the fundamental formulas of loop circuit inductance [25,26]. As for the earth return loop circuit, considering that the earth is a part of the loop circuit, the inductance coefficients are calculated with the Carson formula [27,28].

Deduction for Space 2
The Table 1, Equation (11) are given.

However, the result of
According to the transformer radio transformer of AT in Space 2, the voltage drop between T and N, i.e., u in Figure 4 should be equal to the voltage drop between N and F, i.e., u in Figure 4.
Subsequently, by substituting the obtained values of k1′, k2′ ... k20′ into Equation (15), Δu1, Δu2 ... Δu20 are deduced. Similarly, with Space 1, for the reason that the deduced Δu1, Δu2 ... Δu20 are mathematical expressions of i0′ and Δu1 = … = Δu9 = Δu11 = … = Δu19 while Δu10 = Δu20, we assume that Δu1= … = Δu9 = Δu11 = … = Δu19 = f2(i0′) and Δu10 = Δu20 = g2(i0′). The unit-length electric field return impedance of Space 2 (marked as Z1′) is calculated by Z1′ = f2(i0′)/i0′. The unit-length magnetic field conversion return impedance of Space 2 (marked as Z2′) is calculated by Z2′ = g2(i0′)/i0′. However, the result of (4) is valid only when the spatial position between T and N is symmetrical with the spatial position between N and F and the self-impedance of T is equal to the self-impedance of F. The aforementioned deduced values of k1′, k2′ ... k20′, Z1′ and Z2′ are inaccurate and they can only be regarded as initial results. According to the transformer radio transformer of AT in Space 2, the voltage drop between T and N, i.e., u in Figure 4 should be equal to the voltage drop between N and F, i.e., u′ in Figure 4. For the Figure 4, the values of Z1 and Z2 have been obtained after introducing the iR2 = 2iF1 of Equation (4) into the inductance matrix of Space 1; the values of Z1′ and Z2′ have been obtained after introducing the the inductance matrix of Space 2. According to Equation (4) and the circuit of the AT system, the relation of iT1, iT2, iF1 satisfies For the Figure 4, the values of Z 1 and Z 2 have been obtained after introducing the i R2 = 2i F1 of Equation (4) into the inductance matrix of Space 1; the values of Z 1 and Z 2 have been obtained after introducing the i R1 = 2 D x − 1 · i F1 of Equation (4) into the inductance matrix of Space 2. According to Equation (4) and the circuit of the AT system, the relation of i T1 , i T2 , i F1 satisfies By introducing Equation (17) into , determining x = x 0 (according to the location of load) and setting i F1 as an arbitrary constant of c, it can be found that In Equation (18), u and u are always unequal for the reason that the granted results of are valid only as an approximation. In order to obtain exact results, an iterative calculation method aimed at satisfying the constraint condition of Equation (16) is proposed, as demonstrated in Figure 5. In the beginning, the i R1 /i F1 is adjusted to deduce new values of k 1 , k 2 ... k 20 , Z 1 and Z 2 . On this basis, new u and u are obtained after introducing new i T1 /i T2 and new values of Z 1 and Z 2 into to be adjusted and corresponding calculation is repeated again. Only when u nearly approximates to u' are the deduced values of k 1 , k 2 ... k 20 , Z 1 and Z 2 considered as final and accurate results for Space 2.
When the final and accurate values of Z 1 and Z 2 are determined, the impedance of circuit 1 in Figure 4 (marked as Z a ) and the impedance of circuit 2 in Figure 4 (marked as Z b ) are, respectively, calculated by Equations (19) and (20).
The impedance of the segment loop circuit (marked as ZA) is the parallel impedance of Z a and Z b , namely, When the load is located at the path between the TSS and ATS, the unit-length traction network impedance (marked as Z) is obtained by utilizing Equation (22).
When the load is located at the path between adjacent ATSs, the traction network impedance is composed of Z A and impedance of the long loop circuit (marked as Z B ). As mentioned in Section 2, when x = D, the return current is fully borne by F. The current distribution and impedance of the segment loop circuit are equal to that of the long loop circuit. Therefore, after substituting x = D for x = x 0 in Equation (21), Z B is deduced as So, when the load is located at the path between adjacent ATSs, Z is obtained by utilizing Equation (24).

Experimental Verification
To prove the effectiveness of proposed method effectively and accurately, two short-circuit experiments at different Chinese HSR lines were performed. The first experiment was carried out in the feeding arm of Jiangyou substation on the Chengdu-Chongqing HSR line on 11 October 2015. The second short-circuit experiment was carried out in the feeding arm of Qinglinkou substation on the Xian-Chengdu HSR line on 12 July 2017. The cross section and parameters of traction network conductors in the experiments are shown in Figure 2 and Table 2. The numbers of all the loop circuits in the experiments are listed in Table 1. In the two short-circuit experiments, flexible copper stranded conductor was used to short-circuit the rail and the catenary. Table 2. Parameters of all the conductor wires in the HSR line from Chengdu to Chongqing.  Figure 6, whether for the first short-circuit experiment or the second short-circuit experiment, switches A1, A2, A4, A6, A13, A14, A17 and A18 are closed and the other switches are opened to simulate the standard 2 × 25-kV AT double track power supply mode. The effective values of all the measured currents are shown in Table 3. In addition, the low-voltage side bus voltages of the traction substation and the total currents of the transmission conductors are, respectively, collected at the secondary side of the traction transformer and current transformer in the position of switch A1. According to the collection results, the measured integrated impedance can be obtained.

Cable and Its Horizontal and
Based on the proposed deduction method, the current distribution coefficients of conductors in the short-circuit experiments are calculated in Table 4, after the initial current distribution is continuously adjusted to satisfy the constraint condition of real circuit voltages as much as possible. Compared with existing analysis results of current distribution in the traction network ( [21,29]), the deduced results in Table 4 are reasonable. Furthermore, according to Table 4, the deduced current distribution coefficients of T, N and F are 0.6235, 0.2138 and 0.3765, respectively. According to Table 3 Table 4, after the initial current distribution is continuously adjusted to satisfy the constraint condition of real circuit voltages as much as possible. Compared with existing analysis results of current distribution in the traction network ( [21,29]), the deduced results in Table 4 are reasonable. Furthermore, according to Table 4, the deduced current distribution coefficients of T, N and F are 0.6235, 0.2138 and 0.3765, respectively. According to Table  3 Table 4, after the initial current distribution is continuously adjusted to satisfy the constraint condition of real circuit voltages as much as possible. Compared with existing analysis results of current distribution in the traction network ( [21,29]), the deduced results in Table 4 are reasonable. Furthermore, according to Table 4, the deduced current distribution coefficients of T, N and F are 0.6235, 0.2138 and 0.3765, respectively. According to Table  3 Table 4, after the initial current distribution is continuously adjusted to satisfy the constraint condition of real circuit voltages as much as possible. Compared with existing analysis results of current distribution in the traction network ( [21,29]), the deduced results in Table 4 are reasonable. Furthermore, according to Table 4, the deduced current distribution coefficients of T, N and F are 0.6235, 0.2138 and 0.3765, respectively. According to Table  3     Furthermore, according to measurement records, the unit-length amplitude and phase angle of integrated impedance were 0.3446 Ω/km and 73.54 • in the first short-circuit experiment; and the unit-length amplitude and phase angle of integrated impedance were 0.3614 Ω/km and 71.63 • in the second short-circuit experiment. Relatively, the derived unit-length amplitude and phase angle of integrated impedance were 0.3576 Ω/km and 69.95 • , considering the iterative solution of approaching the constraint condition. Figure 9 depicts the comparison of the above results by using histograms. It can be demonstrated that both the measurement results of the two short-circuit experiments are almost identical to the derived results.  Furthermore, according to measurement records, the unit-length amplitude and phase angle of integrated impedance were 0.3446 Ω/km and 73.54° in the first short-circuit experiment; and the unitlength amplitude and phase angle of integrated impedance were 0.3614 Ω/km and 71.63° in the second short-circuit experiment. Relatively, the derived unit-length amplitude and phase angle of integrated impedance were 0.3576 Ω/km and 69.95°, considering the iterative solution of approaching the constraint condition. Figure 9 depicts the comparison of the above results by using histograms. It can be demonstrated that both the measurement results of the two short-circuit experiments are almost identical to the derived results.

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.