Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis

The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods, and model order reduction (MOR) methods have proven to be very effective in combating such high complexity. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the circuit under study as a function of design parameters such as geometrical and substrate features. Traditional MOR techniques perform order reduction only with respect to frequency, and therefore the computation of a new electromagnetic model and the corresponding reduced model are needed each time a design parameter is modified, reducing the CPU efficiency. Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as geometrical layout or substrate characteristics. We propose a novel PMOR technique applicable to PEEC analysis which is based on a parameterization process of matrices generated by the PEEC method and the projection subspace generated by a passivity-preserving MOR method. The proposed PMOR technique guarantees overall stability and passivity of parameterized reduced order models over a user-defined range of design parameter values. Pertinent numerical examples validate the proposed PMOR approach

Pade-Type Model Reduction of Second-Order and Higher-Order Linear Dynamical Systems

A standard approach to reduced-order modeling of higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for reduced-order modeling of first-order systems. While this approach results in reduced-order models that are characterized as Pade-type or even true Pade approximants of the system's transfer function, in general, these models do not preserve the form of the original higher-order system. In this paper, we present a new approach to reduced-order modeling of higher-order systems based on projections onto suitably partitioned Krylov basis matrices that are obtained by applying Krylov-subspace techniques to an equivalent first-order system. We show that the resulting reduced-order models preserve the form of the original higher-order system. While the resulting reduced-order models are no longer optimal in the Pade sense, we show that they still satisfy a Pade-type approximation property. We also introduce the notion of Hermitian higher-order linear dynamical systems, and we establish an enhanced Pade-type approximation property in the Hermitian case

Interpolation-based parameterized model order reduction of delayed systems

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method 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

A Perturbation Scheme for Passivity Verification and Enforcement of Parameterized Macromodels

This paper presents an algorithm for checking and enforcing passivity of behavioral reduced-order macromodels of LTI systems, whose frequency-domain (scattering) responses depend on external parameters. Such models, which are typically extracted from sampled input-output responses obtained from numerical solution of first-principle physical models, usually expressed as Partial Differential Equations, prove extremely useful in design flows, since they allow optimization, what-if or sensitivity analyses, and design centering. Starting from an implicit parameterization of both poles and residues of the model, as resulting from well-known model identification schemes based on the Generalized Sanathanan-Koerner iteration, we construct a parameter-dependent Skew-Hamiltonian/Hamiltonian matrix pencil. The iterative extraction of purely imaginary eigenvalues ot fhe pencil, combined with an adaptive sampling scheme in the parameter space, is able to identify all regions in the frequency-parameter plane where local passivity violations occur. Then, a singular value perturbation scheme is setup to iteratively correct the model coefficients, until all local passivity violations are eliminated. The final result is a corrected model, which is uniformly passive throughout the parameter range. Several numerical examples denomstrate the effectiveness of the proposed approach.Comment: Submitted to the IEEE Transactions on Components, Packaging and Manufacturing Technology on 13-Apr-201

A Parameterization Scheme for Lossy Transmission Line Macromodels with Application to High Speed Interconnects in Mobile Devices

We introduce a novel parameterization scheme based on the generalized method of characteristics (MoC) formacromodels of transmission-line structures having a cross section depending on several free geometrical and material parameters. This situation is common in early design stages, when the physical structures still have to be finalized and optimized under signal integrity and electromagnetic compatibility constraints. The topology of the adopted line macromodels has been demonstrated to guarantee excellent accuracy and efficiency. The key factors are propagation delay extraction and rational approximations, which intrinsically lead to a SPICE-compatible macromodel stamp. We introduce a scheme that parameterizes this stamp as a function of geometrical and material parameters such as conductor-width and separation, dielectric thickness, and permettivity. The parameterization is performed via multidimensional interpolation of the residue matrices in the rational approximation of characteristic admittance and propagation operators. A significant advantage of this approach consists of the possibility of efficiently utilizing the MoC methodology in an optimization scheme and eventually helping the design of interconnects.We apply the proposed scheme to flexible printed interconnects that are typically found in portable devices having moving parts. Several validations demonstrate the effectiveness of the approac

Compact and accurate models of large single-wall carbon-nanotube interconnects

Single-wall carbon nanotubes (SWCNTs) have been proposed for very large scale integration interconnect applications and their modeling is carried out using the multiconductor transmission line (MTL) formulation. Their time-domain analysis has some simulation issues related to the high number of SWCNTs within each bundle, which results in a highly complex model and loss of accuracy in the case of long interconnects. In recent years, several techniques have been proposed to reduce the complexity of the model whose accuracy decreases as the interconnection length increases. This paper presents a rigorous new technique to generate accurate reduced-order models of large SWCNT interconnects. The frequency response of the MTL is computed by using the spectral form of the dyadic Green's function of the 1-D propagation problem and the model complexity is reduced using rational-model identification techniques. The proposed approach is validated by numerical results involving hundreds of SWCNTs, which confirm its capability of reducing the complexity of the model, while preserving accuracy over a wide frequency range

On the Generation of Large Passive Macromodels for Complex Interconnect Structures

This paper addresses some issues related to the passivity of interconnect macromodels computed from measured or simulated port responses. The generation of such macromodels is usually performed via suitable least squares fitting algorithms. When the number of ports and the dynamic order of the macromodel is large, the inclusion of passivity constraints in the fitting process is cumbersome and results in excessive computational and storage requirements. Therefore, we consider in this work a post-processing approach for passivity enforcement, aimed at the detection and compensation of passivity violations without compromising the model accuracy. Two complementary issues are addressed. First, we consider the enforcement of asymptotic passivity at high frequencies based on the perturbation of the direct coupling term in the transfer matrix. We show how potential problems may arise when off-band poles are present in the model. Second, the enforcement of uniform passivity throughout the entire frequency axis is performed via an iterative perturbation scheme on the purely imaginary eigenvalues of associated Hamiltonian matrices. A special formulation of this spectral perturbation using possibly large but sparse matrices allows the passivity compensation to be performed at a cost which scales only linearly with the order of the system. This formulation involves a restarted Arnoldi iteration combined with a complex frequency hopping algorithm for the selective computation of the imaginary eigenvalues to be perturbed. Some examples of interconnect models are used to illustrate the performance of the proposed technique

Low Power Processor Architectures and Contemporary Techniques for Power Optimization – A Review

The technological evolution has increased the number of transistors for a given die area significantly and increased the switching speed from few MHz to GHz range. Such inversely proportional decline in size and boost in performance consequently demands shrinking of supply voltage and effective power dissipation in chips with millions of transistors. This has triggered substantial amount of research in power reduction techniques into almost every aspect of the chip and particularly the processor cores contained in the chip. This paper presents an overview of techniques for achieving the power efficiency mainly at the processor core level but also visits related domains such as buses and memories. There are various processor parameters and features such as supply voltage, clock frequency, cache and pipelining which can be optimized to reduce the power consumption of the processor. This paper discusses various ways in which these parameters can be optimized. Also, emerging power efficient processor architectures are overviewed and research activities are discussed which should help reader identify how these factors in a processor contribute to power consumption. Some of these concepts have been already established whereas others are still active research areas. © 2009 ACADEMY PUBLISHER