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On the performance of weighting schemes for passivity enforcement of delayed rational macromodels of long interconnects
Abstract
This paper focuses on robust black-box macromodeling of electrically long interconnects. These structures can be represented by very compact and efficient models that combine in their constitutive equations both closed-form delay operators and low-order rational coefficients. We present a simple perturbation approach for efficient passivity enforcement of these delayed rational macromodels, and we show how suitable norm weighting schemes may significantly improve in-band accuracy preservation.
... The integrals in (17) can be evaluated analytically by solving suitably defined Lyapunov equations [12]. Equation (16) can be further simplified by collecting submatrices W ...
... The integrals in (17) can be evaluated analytically by solving suitably defined Lyapunov equations [12]. Equation (16) can be further simplified by collecting submatrices W i,j m,n in a block-matrix W i,j , leading to ...
This paper describes a robust and accurate blackbox macromodeling technique, in which the constitutive equations combine both closed-form delay operators and low-order rational coefficients. These models describe efficiently electrically long interconnect links. The algorithm is based on an iterative weighted least-squares process and can be interpreted as a generalization of the well-known Vector Fitting. The paper is focused, in particular, on the passivity enforcement of these models. We present two perturbation methods and we show how the accuracy of the models is well-preserved during the passivity enforcement process.
... where the minimum is taken over the unknown delays τ i,j m and matrix rational functions Q i,j m (s) is performed using the so-called Delayed Vector Fitting (DVF) [7], which is well documented and is not further commented here. We remark that also the passivity of (1) is easily checked and enforced, see [10], [12]. ...
We present a Waveform Relaxation approach for fast transient simulation of electrically long high-speed channels. The proposed technique is based on a two-level transverse and longitudinal partition of the coupled interconnect. All couplings and feedback from terminations are represented as correction sources, which become explicit in a Waveform Relaxation framework. We show that the proposed schemes are consistent and allow for very efficient implementations. Simulation results show speedup factors of up to two orders of magnitude with respect to circuit-based (SPICE) solvers.
... , L through Delayed Vector Fitting (DVF) or the Delayed Sanathanan-Koerner (DSK) iterations, see [5] [4]. Model passivity can also be checked end enforced [6] [7]. The main advantage of DRM operators is that their time-domain application can be cast as a delayed recursive convolution, whose numerical evaluation has a cost that scales only linearly with the number of time steps to be computed [10]. ...
This paper presents a fast simulation method for long coupled multi-chip interconnects. The channel is represented through a time-domain passive delay-rational macromodel, which is identified from tabulated scattering samples. A two-level Waveform Relaxation (WR) framework is then applied in order to perform fast transient simulation of the terminated channel via an iterative process. A new over-relaxation scheme is introduced for improving the convergence of the WR iterations.
... An alternative and purely algebraic characterization of this impulse perturbation energy is obtained as illustrated in [12], obtaining ...
This paper presents a general strategy for the electrical performance assessment of electrically long multi-chip links. A black-box time-domain macromodel is first derived from tabulated frequency responses in scattering form. This model is structured as a combination of ideal delay terms with frequency-dependent rational coefficients. Two passivity enforcement schemes are then presented, enabling safe and reliable transient simulations of the multi-chip link with a standard circuit solver, including nonlinear drivers and receivers.
Flexible structure dynamics with collocated force actuators and position sensors lead to negative imaginary (NI) systems. However, in some cases, the models obtained for these systems may not satisfy the NI property. This paper provides a new method for enforcing such models to be NI. The results are based on a study of the spectral properties of related Hamiltonian matrices. A test for the negativity of the imaginary part of a corresponding transfer function matrix is first performed by checking for the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues, the system is not NI. In such cases, a first-order perturbation is presented for the precise characterization of frequency bands where violations of the NI property occur. This characterization is then used for the design of an iterative perturbation scheme for state matrices aimed at displacing the imaginary eigenvalues of the Hamiltonian matrix away from the imaginary axis.
This paper presents a waveform relaxation approach for efficient transient simulation of fully-coupled high-speed channels of medium to large electrical length. The channel is first described via a passive Delayed Rational Macromodel, which is suitably identified from tabulated scattering data. The structure of the macromodel is then exploited to setup a two-level iterative relaxation scheme for the solution of terminal voltage and current waveforms. Numerical results show that the proposed scheme outperforms standard SPICE-based solvers at no loss of accuracy.
This paper presents a class of numerical schemes for fast transient simulation of electrically long and complex linear channels terminated by linear or nonlinear networks. The common denominator of all schemes is waveform relaxation. Domain decomposition approaches based on longitudinal and transverse partitioning are pursued, leading to various iterative methods characterized by different properties and numerical efficiency. For each scheme, we present a detailed convergence analysis and a set of numerical results obtained on industrial benchmarks. The main contribution of this paper is a novel theoretical framework based on a combination of longitudinal and transverse relaxation, leading to particularly efficient simulations when combined with compact channel representations based on delay-rational macromodels. Our prototypal implementation outperforms SPICE solvers, with speedup of up to two orders of magnitude.
This paper presents a general strategy for the electrical performance and signal integrity assessment of electrically long multichip links. A black-box time-domain macromodel is first derived from tabulated frequency responses in scattering form. This model is structured as a combination of ideal delay terms with frequency-dependent rational coefficients. A new identification scheme is presented, which is based on an initial blind delay estimation process followed by a refinement loop based on an iterative delayed vector fitting process. Two alternative passivity enforcement schemes based on local perturbations are then presented. The result is an accurate and guaranteed passive delay-based macromodel, which is synthesized as a SPICE-compatible netlist for channel analysis. The proposed procedure enables safe and reliable circuit-based transient simulations of complex multichip links, including nonlinear drivers and receivers. The performance of the proposed flow is demonstrated on a large number of channel benchmarks.
In this paper, a generalized theory for passivity verification of delayed rational function (DRF) macromodels representing electrically long networks that are characterized by multiport tabulated scattering parameters is presented. In the proposed approach, passivity verification of DRF macromodels is formulated as a quasi-periodic frequency-dependent generalized eigenvalue problem, using which, the necessary search region for passivity violations is reduced to just a single period along the imaginary axis. Necessary theoretical foundations and the related proofs are developed. Further, a computationally more efficient method based on half-Hamiltonian size frequency-dependent generalized eigenvalue problem is developed. Numerical validations for both the full-size and half-size formulations are also presented.
In this paper, an efficient passivity verification algorithm is presented for delayed rational function based macromodels obtained from tabulated scattering parameter data. The proposed approach is based on a new half-size frequency dependent generalized eigenvalue problem, which reduces the necessary search region to a single finite interval along the imaginary axis. Numerical results validating improved efficiency of the proposed algorithm over previous techniques are also presented.
We present a robust and efficient scheme for the generation of delay-based macromodels from frequency-domain tabulated responses. These responses can come from both simulation or direct measurement. The main algorithm is based on an iterative weighted least-squares process for the identification of delayed rational approximations. In case that pole relocation is performed during the iterations, the scheme can be interpreted as a generalization of the well-known vector fitting algorithm to delayed systems. Therefore, we denote this algorithm as delayed vector fitting (DVF). In case no pole relocation is performed, the scheme generalizes the so-called Sanathanan-Koerner iteration, calling for the denomination of delayed Sanathanan-Koerner algorithm. These techniques produce compact macromodels that are readily synthesized in SPICE-compatible equivalent circuits including delayed sources or ideal transmission line elements. These macromodels outperform classical rational macromodels in terms of simulation time. Several examples illustrate the advantages of proposed methodology.
This paper introduces two different algorithms for passivity enforcement of transmission-line macromodels based on the generalized method of characteristics (MoC). The first scheme corrects passivity violations via perturbation of purely imaginary solutions of a nonlinear Hamiltonian eigenvalue problem. The second scheme is based on a linearly constrained quadratic optimization formulated at carefully selected frequency samples. Both schemes can be viewed as extensions to the case of MoC-based macromodels of existing techniques that were only available for lumped macromodels. The resulting passive delay-based macromodels can be safely employed in system-level signal integrity simulations, due to their inability to produce any spurious energy gain. Several numerical examples illustrate the behavior of both schemes on transmission line structures of practical interest, highlighting their main differences and the open problems deserving further research work.
Modern packaging design requires extensive signal integrity simulations in order to assess the electrical performance of the system. The feasibility of such simulations is granted only when accurate and efficient models are available for all system parts and components having a significant influence on the signals. Unfortunately, model derivation is still a challenging task, despite the extensive research that has been devoted to this topic. In fact, it is a common experience that modeling or simulation tasks sometimes fail, often without a clear understanding of the main reason. This paper presents the fundamental properties of causality, stability, and passivity that electrical interconnect models must satisfy in order to be physically consistent. All basic definitions are reviewed in time domain, Laplace domain, and frequency domain, and all significant interrelations between these properties are outlined. This background material is used to interpret several common situations where either model derivation or model use in a computer-aided design environment fails dramatically. We show that the root cause for these difficulties can always be traced back to the lack of stability, causality, or passivity in the data providing the structure characterization and/or in the model itself.
This paper introduces a new macromodeling algorithm for electrically long interconnects. This technique starts from frequency-domain scattering data and produces compact macromodels based on multiple delay extraction and rational approximations. The proposed model structure significantly reduces model complexity and simulation time with respect to standard purely rational macromodels.
Passivity has become a critical issue while modeling high-speed networks characterized by tabulated data. For modules containing long delays, macromodeling based on delayed rational functions has been gaining popularity in recent years. However, there are currently no passivity verification or enforcement methods available for such macromodels. To address this issue, this paper proposes passivity verification and enforcement algorithms for delayed rational function macromodels of high-speed modules, characterized by tabulated data. Necessary theoretical foundation and validating examples are presented.
We introduce a new macromodeling scheme for electrically long interconnects characterized by tabulated frequency responses. The transfer function of the interconnect is modeled as a superposition of multiple single-delay atoms, which are identified via a selective inversion of a Gabor transform of the raw frequency data. Each atom is then approximated by a delayed rational function, leading to a highly-efficient SPICE-ready macromodel
This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in state-space form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system input-output transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on first-order perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that first-order perturbation is applicable. Several examples illustrate and validate the procedure.
This letter introduces a new method for compact macromodeling of high-speed circuits with long delays, characterized by tabulated time-domain data. The algorithm is based on partitioning the response and subsequently approximating each partition with a low-order sum-of-exponentials, delayed in time-domain. This results in a compact low-order macromodel in the form of delayed-differential equations, which can be efficiently analyzed using SPICE like simulators.
This paper presents a generalized theory of passivity verification in delay-based macromodels for multiconductor transmission line networks generated using the method of characteristics (MoCs). We demonstrate that the passivity in an MoC macro-model is equivalent to the nonnegative definiteness in the admittance matrices of two submodels. We then provide the necessary and sufficient conditions for each submodel to have a nonnegative definite admittance matrix. The presented theory develops an algebraic test to verify the passivity in MoC macromodels. Numerical results demonstrate the validity of the proposed theory.
The paper describes a general methodology for the fitting of
measured or calculated frequency domain responses with rational function
approximations. This is achieved by replacing a set of starting poles
with an improved set of poles via a scaling procedure. A previous paper
(Gustavsen et al., 1997) described the application of the method to
smooth functions using real starting poles. This paper extends the
method to functions with a high number of resonance peaks by allowing
complex starting poles. Fundamental properties of the method are
discussed and details of its practical implementation are described. The
method is demonstrated to be very suitable for fitting network
equivalents and transformer responses. The computer code is in the
public domain, available from the first author