Mathematical Model of Voltage and Current Distribution in Transmission Line Using DSVs and Exponential Approximation

In this paper a mathematical model is developed for the distribution of voltage and current in the transmission line using distributed state variables. It gives the time domain solution of the terminal voltage and current as well as their line distributions. This is achieved by treating voltage and current distributions as distributed state variables (DSVs) and turning the transmission line equation into an ordinary differential equation. Overall the transmission line is treated like other lumped dynamic components, such as capacitors, inductors, etc using backward differentiation formula for time discretization, the DSV transmission line component is converted to a simple time domain companion model, from which its local truncation error can be derived. As the voltage and current distribution get more complicated with time, a new piecewise exponential with controlled accuracy is invented. A segmentation algorithm is also devised so that the line is dynamically bisected to guarantee that the total piecewise exponential error is a small fraction of the local truncation error. Using this approach, the user can see the line voltage and current at any point and time freely without explicitly segmenting the line before starting the simulation.