Parameterized model order reduction of delayed systems using an interpolation approach with amplitude and frequency scaling coefficients

Abstract

When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. We present an innovative PMOR technique for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, amplitude and frequency scaling coefficients and positive interpolation schemes. It is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach. IntroductionComplex high-speed systems require 3-D electromagnetic (EM) methods [1,2] as analysis and design tools. Large systems of equations are usually generated by the use of these methods and model order reduction (MOR) techniques are crucial to reduce the complexity of EM models and the computational cost of the simulations, while retaining the important physical features of the original system [3,4]. Over the last years, the development of methods to build reduced order models (ROMs) of EM systems has been intensively investigated, with applications to interconnects, vias and high-speed packages [5,6].When signal waveform rise times decrease and the corresponding frequency content increases or the geometric dimensions become electrically large, time delays must be taken into account and included in the modeling. Simply using rational models can result in significant errors and artifacts. Therefore, systems of neutral delayed differential equations (NDDE) [7] becomes fundamental to accurately describe the behavior of such structures.Traditional MOR techniques perform model reduction only with respect to frequency, but a typical design process includes design space optimization and exploration, and thus it requires repeated simulations for different design parameter values (e.g. geometrical layout or substrate characteristics). Such design activities call for parameterized model order reduction (PMOR) methods that can reduce large systems of equations with respect to frequency and other design parameters.Over the years, a limited number of PMOR methods for NDDE systems have been developed [8][9][10]. They are mainly