Wideband Macromodels in Finite Element Method

Frequency domain behavior of S‐parameters piecewise‐linear fitting in a digital‐wave framework

“…Hence, a PWC model of the impulse response is equivalent in the frequency domain to a superposition of delayed sinc functions, whose phase is shifted by a term equal to e ÀjωD i for i ¼ 1, …, N À 1. The extension to general multiport systems is straightforward: each element of the impulse response matrix can be represented via ( 4)- (6).…”

“…The RectSinc transformation defined by (6) gives the exact spectrum of an impulse response composed by a sum of delayed rectangles, obtained from the time derivative of the corresponding step response interpolated by a piecewise linear function of time. It can be applied both to the extracted models and to input data containing the sampled step response.…”

“…It can be applied both to the extracted models and to input data containing the sampled step response. The computation of spectrum (6) during the model extraction process will be described in Section 4, while the estimation of the reference spectrum related to the input data will be discussed in the following. In this case, the time intervals ΔT i are all equal to the T IN sampling interval of the N s input samples.…”

“…In the first case, it can be obtained through measurements of a vector network analyzer (VNA) or by numerical simulations using full-wave field or circuit solvers, while time-domain reflectometry (TDR/TDT) and transient field or circuit solvers can be adopted in the latter case. While there is a vast literature for macromodeling techniques using frequency-domain data, [1][2][3][4][5][6][7][8][9][10] only few techniques have been developed over the years for time-domain macromodeling. For example, the Time-Domain Vector Fitting (TD-VF) technique is introduced in References 11,12 for the identification of the dominant poles of the structure using raw data transient excitations and responses at the ports of the structure.…”