Transient Simulation of Voltage and Current Distributions within Transmission Lines

The problem of analyzing transient in transmission line circuits is studied with emphasis on obtaining the transient voltage and current distributions. A new method for solving Telegrapher equation that characterizes the uniform transmission lines is presented. It not only gives the time domain solution of the line terminal voltage and current, but also their distributions within the lines. The method achieves its goal by treating the voltage and current distributions as distributed state variables and transforms the Telegrapher equation into an ordinary differential equation. This allows the coupled transmission lines to be treated as a single component that behaves like other lumped dynamic components, such as capacitors and inductors. Using Backward Differentiation Formulae for time discretization, the transmission line component is converted to its time domain companion model, from which its local truncation error for time step control can be derived. As the shapes of the voltage and current distributions get more complicated with time, they can be approximated by piecewise exponential functions with controllable accuracy. A segmentation algorithm is thus devised so that the line is dynamically bisected to guarantee that the total piecewise exponential approximation error is only 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 segment the line before starting the simulation.