Voltage response of distributed RC lines with a discontinuous initial potential distribution

Abstract

Prior work by this author has shown how the voltage step response of an initially quiescent RC line can be concisely expressed in terms of a set of higher transcendental functions. For convenience, the latter were defined in termsof integrals of elliptic theta functions and are termed “diffusion functions”. These same diffusion functions are also found to be useful in expressing closed-form solutions for the voltage response of distributed RC lines with a non-quiescent and discontinuous initial potential distribution. Results herein provide functional expressions for this voltage response with either open-circuited or short-circuited terminations. A variety of solution plots show how the responses change for different positions of the discontinuity. Also, plots of the diffusion functions are supplied showing their behavior over one complete cycle and more information on their symmetry properties is provided than was included in the initial paper.