Electrodynamics of electric power transmission and losses in devices of electric transport systems

For the first time, the “field” approach for explaining the processes of transmission and generation of electric power losses in devices of electric transport systems is described and theoretically substantiated on the basis of the theory of electromagnetic field. The results of the solution of the system of electromagnetic field equations show that it is energetically appropriate to design low-floor types of electric rolling stock. A qualitative view of electric power flows arriving through the air of the feeder zone from the traction substation and entering to the electric rolling stock through the roof and the front part of its body is presented. It is established that the main flow of energy enters through the roof porcelain insulator. At the same time, the electromagnetic waves partly penetrate into the metal surfaces of roof and frontal part of the body, and partially they are reflected from them creating losses of active power. The results of calculations of these losses, power factor and reactive power factor of the electric locomotive roof are shown. The relation between the standing waves, formed in the feeder zone, and the reactive power consumed by the electric rolling stock is established.


Introduction
Existing scientific publications, highlighting findings of electric power processes in electric transport systems, are based on the "circuit approach", i.e. on the theory of linear or nonlinear electric and magnetic circuits. Though, in essence, all electromagnetic and electric power processes at their full and detailed studying require the "field approach" based on the theory of electromagnetic field. In our opinion, it is nearly the main reason why a set of processes or existing facts in electric power engineering is not explained sufficiently. This "circuit approach" restricts and limits possibilities of scientific management of power processes. In this respect a lot of things are unexplainable. For instance, in case of "leak" (i.e. losses of the electric power in electric traction system) the meter registrations of electric power of traction substations (TS) and electric rolling stock (ERS) differ. These losses are already called "conditional losses" and they are in a directive way regulated by the Main Department of locomotive facilities of Ukrzaliznytsia (Ukrainian Railway). And though, many authors are trying to justify the reasons of occurring and change of these losses, but in our opinion these arguments are not convincing.
And, after all, we present the most important justification of the above. According to the theoretical bases of electrical engineering, electromagnetic energy (electric power) is transmitted from the source (traction substation) to the consumer (electric locomotive) not trough the wires (i.e. electric circuit), but through the dielectric (air) between (and behind) the wires and it is transmitted not by charges, but by electromagnetic field, i.e. by electromagnetic waves.
In this regard, the author of the present work covers a set of electrical power issues concerning power transmission in electric transport systems. For this purpose we will recall that, as it is well-known from the theoretical electrical engineering, the power transmission (and information as well) through dielectric is characterized by the Poynting vector  S in respect of both quantity and direction [1, p. 421], [2, p. 147], [3, p. 220]: is an angular frequency of sinusoidally varying voltage and current which produce electromagnetic field (waves).
From (2), (9) and (10) it follows that if mediums 1 and 2 have identical parameters, the reflected wave is absent. In this case, the only incident wave is in medium 1 and refracted wave takes place in medium 2. Irrespective of the incidence angle of the incident wave on the bounder of mediums 1 and 2, the refracted wave propagates in the conducting medium, first, always in the direction of a normal to the border of mediums and, secondly, this wave dies out at the depth  defined as: And finally the last remark: since waves, being reflected, don't change axes of vectors but can change their sign, instead of the corresponding vectors E  and H   it is possible to consider their vector modules.

Power flow in the air between a catenary system and a rail
Let's consider power transmission from a DC traction substation to an electric locomotive for the case of onesided supply and the most widespread catenary system of 2MF-100+A185 type and the rail of R65 type ( Fig. 1) The electromagnetic field in the dielectric (air) surrounding the catenary system is described by the system of equations [3, pp. 214-215], [5, pp. 29-30]: (18) Expression (13) demonstrates that electric field in the air is eddy-free, i.e. potential; therefore (18) is correct. Accepting that dielectric medium is homogenous, i.e. const  a  , and after inserting (18) in (15), we receive Laplace's equation where the potential V (x, y, z) is subjected in each dielectric point of the traction power system. Let's consider that in the first approximation the potential V is changing only relatively to the coordinate x, i.e. from the catenary system to the rail ( Fig. 1), then (19) should to be simplified: The solution of (20) is possible under desired boundary conditions [5, pp. 37-39] and one of them is the equality of tangential components of electric field intensity ( Fig. 1): in the conductor can be defined from well-known equation: where J is a current density in the catenary system,  is electrical conductivity of material of the system. For example, feeder current at one train, operated by the DE1 electric locomotive, doesn't exceed 2000 [A] I  , and a cross section of the equivalent catenary system of 2MF-100+A185 type has to be found according to [6, p. 52 4.3 10  times. Therefore, calculating the electric field in the air around the catenary system it is possible to ignore component 1t E  and use boundary electrostatics condition, i.e. condition of equipotential surface of a conductor. Hence, the intensity in the air between the catenary system and the rail is defined as The intensity of magnetic field will be found using the Ampere's law ( Fig. 1) as where d is a diameter of the equivalent catenary wire. At a further analysis we will take into account that to the strength ) (x H , determined by (24) and produced by equivalent catenary wire, it is necessary to add the component produced by the current in rails. The equations for resultant intensity may to be found at any point (in two-wire power line), for example, according to [3, pp. 176-181]. Substitute (23) and (24) into (1), in the first approximation we get (without considering H  , produced by the current in a rail) the equation of the absolute value of the Poynting vector  S in dielectric (air) points between the catenary system and the rail: In points 1, 2, 3 and 4 ( Fig. 1) the vector H  is directed behind a flat surface of the sheet as shown in Fig. 1 P by 24.7%, which corresponds to so-called "conditional" losses, "leakage" losses, which are still inexplicable and which are not counted by electricity meters, since their principle of operation is based on the circuit theory. Therefore, the electric power analysis of electric transport systems have be carried out on the basis of the electromagnetic field theory. 4) According to (25), the highest power density per time is transmitted in the space of small values х (Fig. 1), that is, in the space around the contact wires and the rail.
It is possible to show that the same condition regarding the direction of electric power is observed also in the space above the catenary system and under the rail base with the ground; but values of the vector  S will be different, considering non-homogeneity of all traction network.
Considering the above and expression (25), it is possible to make the following strategic proposals regarding devices of electric traction system. According to (25), with the decrease of the value h, i.e. with the decrease of distance between the contact wire and the head of a rail, density of the electric power transmitted , therefore in the projected electric traction systems it is reasonable to reduce h, and construct low-floor (but probably longish) types of ERS. There is nothing irregular in this proposal if we take into account that nowadays one of the most important indicators of technical progress is the reducing the space occupied by the electromagnetic device [7, 8], as it is for a reason microcomputers, electrical micro machines, different micro devices and the like are created. In the context of energetics for production of the mentioned devices it is reasonable to enter the coefficient of utilization of construction space e  : where e 0 1    .

Electric power flow through the roof bushing insulator and through the roof of electric locomotive
Out of the total electric power, flowing (carried by waves) to ERS out of the air space of the traction network it is possible to identify two flows: the main and the minor. The main flow is the energy flow representing the greatest part of the energy received by ERS. It concentrates around traction network wires and enters to electrical devices of the high-voltage cabinet through the roof bushing insulator, i.e. roof surface (Fig. 2) and through the bottom of the cabinet from the space around the rail. The minor flow is the electric power flow carried by waves which fall on the front part of electric locomotive body, which consists, in its turn, of metal and dielectric (cab windscreen) parts (Fig. 2). In the beginning we will analyse the main flow in which it is possible to identify the following components of the Poynting vector carrying it (Fig. 2).  (Fig. 2) enters into the highvoltage cabinet of ERS from the space around the rail (i.e. "from the bottom" of ERS). Wave propagation in this space and power transmission needs more in-depth studying which is not given here yet.
2. The power vector (Poynting) roof  S penetrates into the metal roof of ERS through its surface (Fig. 2). The vector roof  S is directed normally to the surface. Roof material is a sheet structural steel of 2 [mm] thick with parameters: 6 7 10 [Sm/m]    ; r 1000   . Electromagnetic harmonic waves of various frequencies, falling from air 1 onto the border with the surface of metal roof 5 (Fig. 2), are partially reflected, and partially penetrate into the metal of the roof, gradually dying out and at the same time causing losses of electric power in roof metal.
Wave impedance of the roof metal 5 Z , found according to (10) for [

Energy flow falling on the front of the locomotive body
Electromagnetic waves propagating in the middle part of the airspace 1 between the catenary system and the rail, fall onto the front part of the locomotive body which consists of glass 7 and lower metal part 8 (Fig. 2). The thickness of cab windscreen is 15 mm and 10 5 . 5 r7    (let's take 9.0). On the border "air 1glass 7" waves are partially reflected with the coefficient of