Application of Conformal Mappings to Determine the Location of Rolling Stock on a Section of a Rail-Wire Line

The availability of information regarding the position of rolling stock is crucial for the organization of train movements. Existing systems employed on the railways of the Russian Federation exhibit relatively low accuracy in determining the location of rolling stocks. Improvements in the accuracy of position detection can be achieved through modifications to the algorithms for processing information obtained from the rail-wires line. This paper proposes the application of the mathematical framework of conformal mappings to analyze the variations in the areas of input impedances of the rail-wires line across all possible values of the load impedance of that rail-wires line. Using the Spyder scientific environment and the Python programming language, a simulation of a segment of the rail-wires line was conducted, leading to the construction of nomograms for input impedances under various combinations of insulation resistance values of the rail-wires line, the frequency at which the impedance is measured, and the coordinates of the rolling stock's position on the rail-wires line. The simulation results demonstrated that the mathematical framework of conformal mappings provides a clear visualization of the changes in the areas of input impedance for various parameters of the rail-wires line, with the highest informational value being achieved through measurements of input impedance at tonal frequency ranges.