Analytical Solutions for Distributed Interconnect Models—Part I: Step Input Response of Finite and Semi-Infinite Lines

With Fourier series and Fourier integrals, a new and systematic approach, called the Amir and Nasser (AMN) method, is proposed to derive exact analytical expressions for step input response of distributed resistance capacitance (RC), inductance-capacitance (LC), and resistance-inductance-capacitance (RLC) models of interconnects. These solutions are obtained for time-domain responses of any arbitrary point on a finite interconnect line, considering initial voltage through the line. This method is appropriate for both on-chip and printed circuit board wires without any limitations in their length or characteristic parameters. The developed solutions are expressed as infinite summation of sinusoidal terms. An accuracy of over 99.5% is observed for the expressions compared with the HSPICE simulations for at most a number of several tens of sinusoidal terms for Fourier series. It is shown that ignoring the initial voltage through the line leads to considerable error as high as 33% at the far end voltage in global interconnects for 65-nm technology node. The AMN method is extended to semi-infinite distributed RC and RLC interconnects for which exact closed-form expressions are achieved.