Analytical expressions for transient response of a periodic structure consisting of even number of transmission lines'sections capacitively loaded at junctions

Abstract

Some configurations of intgcormections in electmnic gstcms may be modeled by cascaded transmission 1im.s' sections loaded capacitively at junctions. For investigation of high-speed electromagnetic signals propagation in such stmctues a transient analysis using Laplace transform was proposed by Gu and Kong [ 11. The merit of the analysis c0nsist.s in analytical expressions for transmitted and reflected signals fw a complex periodic &ructure consisting of arbitrary number of transmission lines' sections, As a result, a transient response of single line strwhre and nos&alk in two coupled r$firtures may be calculated analytically. The authors of this paper used the formulae of Gu and Kong for transmitted wave accounting t a m s generated through two reflections when the difference of sections' paramem was not very large [Z] and presented recently mcre accurak formulae far transmitted wave accounting tams generated through four [3] and six 141 reflections suitable for widerrange of the structure parameta. Unforhmately, the wiginal fotmulae of Gu and Kong have been derived far case when parameters of external sections rzn& be !. he samg and for case of Q& number (n=3,5,7,. ..) of transmiasion lines' sections only. This restriction makes impossible the calculation of some cafes of structwes taking place in practice. In this paper using approach of Gu and Kong the analytical expressions for transient response f a case when parameters of extanal sections may be diffgent and for case of even number (&,4,6,. . .> of trammission lines' &ions 8tr! presented. A generalized equivalent circuit of periodic structure under consideration is shown in Figl, where YO, Y,,+l, b, %+I, and Y1, Y2, h, are the characteristic admittances and total signal delays of two the same CJrtanal and n internal transmission lines' sections of two different kinds acccrdingly, while cd is value of equivalent lumped capacitances of discontinuities. We consider the transient response of the circuit considered on ramp input signal v,@) = (vd It,. $u(t) -(tt,. p ( ft,. )] , where U(&) is a etq, function, and VI, and t, are the magnitude and the rise time of the ramp input signal. The final analytic expressions for transmitted signal and for reflected signal VI&) have the following forms. %+1,%+1 v -YZIT2