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Passive Reduced Order Macromodeling using Hamiltonian Matrix Pencil Perturbation
Abstract
In this paper, a method for obtaining a reduced order macromodel based on measured parameters is presented. Loewner Matrix based macromodeling may require a higher than optimal system order in order to preserve the passivity of the macromodel. In this paper, a modified order selection scheme, combined with the Hamiltonian Matrix Pencil perturbation methodology is used in order to obtain a reduced order
passive Loewner Matrix based macromodel based on measured scattering parameters. The efficiency of the new approach is shown in the examples.
... Such approaches are practical for cases where a large or distributed model can be found. More recently, a new class of more direct approaches has been proposed in the literature which rely on the numerical generation of a spice-compatible time-domain macromodel directly from frequency domain Sor Y -parameter data that can be computed using full-wave simulation [8]- [23]. This approach is general for any passive structure that can be simulated using full-wave techniques, and the resulting macromodel is compatible with a wide variety of simulators. ...
... The contribution of this paper can be classified in the category of macromodeling methods based on simulated frequency-domain parameters [8]- [23]. The general approach of these methods is to automatically compute an approximation of the system in terms of a pole/residue representation, rational function representation, or descriptor system representation. ...
... In [30] a method was proposed to obtain a reduced but sufficiently accurate Loewner Matrix based macromodel for systems described by Y -parameters. A similar approach was proposed in [23] for systems described by S-parameters. In both cases, this was achieved by relaxing the passivity requirement during the order selection process, and then enforcing passivity using a suitable algorithm for perturbing the system. ...
Loewner Matrix interpolation based macromodeling methods can generate an accurate macromodel for large and complex structures. However, there is less flexibility in the choice of the macromodel order due to the need to ensure passivity. In this paper, an algorithm is proposed to obtain a reduced order model based on Loewner Matrix interpolation while still maintaining the passivity of the macromodel. This is achieved by first relaxing the passivity requirement during the order selection process, then a new passivity enforcement scheme with significantly improved speed and convergence properties is applied. The proposed approach results in a significantly reduced macromodel size while maintaining accuracy and passivity conditions. The
efficiency of the new approach is shown in the examples.
Macromodeling techniques using Loewner Matrix (LM) interpolation were proposed recently as a way to generate time-domain macromodels based on simulated Y-parameters. These approaches scale very well with respect to the number of ports as well as number of poles in the system. However, these methods become less efficient in terms of accuracy and passivity for Y-parameters obtained using Electromagnetic (EM) simulators.
In this paper, we propose a LM based interpolation technique that is applicable for large scale distributed systems described by full-wave Y-parameters. An algorithm to approximate and extract the port impedance matrix, D, directly from the data is proposed. Additionally, an order selection scheme is proposed that results in an accurate macromodel while maintaining passivity. The efficiency and accuracy of the proposed approach is illustrated using comparisons to a standard technique.
With the rapid developments in very large-scale integration (VLSI)
technology, design and computer-aided design (CAD) techniques, at both
the chip and package level, the operating frequencies are fast reaching
the vicinity of gigahertz and switching times are getting to the
subnanosecond levels. The ever increasing quest for high-speed
applications is placing higher demands on interconnect performance and
highlighted the previously negligible effects of interconnects such as
ringing, signal delay, distortion, reflections, and crosstalk. In this
review paper various high-speed interconnect effects are briefly
discussed. In addition, recent advances in transmission line
macromodeling techniques are presented. Also, simulation of high-speed
interconnects using model-reduction-based algorithms is discussed in
detail
Recently, Loewner Matrix (LM) method was introduced for generating time-domain macromodels based on frequency-domain measured parameters. For the case of lumped systems the order is clearly defined and is detected easily by the LM algorithm. However, for the case of distributed systems with a theoretically infinite number of poles, an optimum order is not easily obtained. In this paper we propose an order selection method for distributed systems defined by simulated or measured S-parameters. The advantages of the proposed approach are illustrated using numerical examples.
Recently, Loewner matrix (LM)-based methods were introduced for generating time-domain macromodels based on frequency-domain measured parameters. These methods were shown to be very efficient and accurate for lumped systems with a large number of ports; however, they were not suitable for distributed transmission-line networks. In this paper, an LM-based approach is proposed for modeling distributed networks. The new method was shown to be efficient and accurate for large-scale distributed networks.
Passivity is an important property of circuits and systems to guarantee stable global simulation. Nonetheless, nonpassive models may result from passive underlying structures due to numerical or measurement error/inaccuracy. A postprocessing passivity enforcement algorithm is therefore desirable to perturb the model to be passive under a controlled error. However, previous literature only reports such passivity enforcement algorithms for pole-residue models and regular systems (RSs). In this paper, passivity enforcement algorithms for descriptor systems (DSs, a superset of RSs) with possibly singular direct term (specifically, or ) are proposed. The proposed algorithms cover all kinds of state-space models (RSs or DSs, with direct terms being singular or nonsingular, in the immittance or scattering representation) and thus have a much wider application scope than existing algorithms. The passivity enforcement is reduced to two standard optimization problems that can be solved efficiently. The objective functions in both optimization problems are the error functions, hence perturbed models with adequate accuracy can be obtained. Numerical examples then verify the efficiency and robustness of the proposed algorithms.
Passivity is a crucial property of macromodels to guarantee stable global (interconnected) simulation. However, weakly nonpassive models may be generated for passive circuits and systems in various contexts, such as data fitting, model order reduction (MOR) and electromagnetic (EM) macromodeling. Therefore, a post-processing passivity enforcement algorithm is desired. Most existing algorithms are designed to handle pole-residue models. The few algorithms for state space models only handle regular systems (RSs) with a nonsingular D+DT term. To the authors' best knowledge, no algorithm has been proposed to enforce passivity for more general descriptor systems (DSs) and state space models with singular D+DT terms. In this paper, a new post-processing passivity enforcement algorithm based on perturbation of Hamiltonian-symplectic matrix pencil, PEDS, is proposed. PEDS, for the first time, can enforce passivity for DSs. It can also handle all kinds of state space models (both RSs and DSs) with singular D+DT terms. Moreover, a criterion to control the error of perturbation is devised, with which the optimal passive models with the best accuracy can be obtained. Numerical examples then verify that PEDS is efficient, robust and relatively cheap for passivity enforcement of DSs with mild passivity violations.
An efficient passivity test based on canonical projector techniques is proposed for descriptor systems (DSs) widely encountered in circuit and system modeling. The test features a natural flow that first evaluates the index of a DS, followed by possible decoupling into its proper and improper subsystems. Explicit state-space formulations for respective subsystems are derived to facilitate further processing such as model order reduction and/or passivity enforcement. Efficient projector construction and a fast generalized Hamiltonian test for the proper-part passivity are also elaborated. Numerical examples then confirm the superiority of the proposed method over existing passivity tests for DSs based on linear matrix inequalities or skew-Hamiltonian/Hamiltonian matrix pencils.
This paper addresses the problem of modeling systems from measurements of their frequency response. For multiport devices, currently available techniques are expensive. We propose a new approach which is based on a system-theoretic tool, the Loewner matrix pencil constructed in the context of tangential interpolation. Several implementations are presented. They are fast, accurate, they build low order models and are especially designed for a large number of terminals. Moreover, they identify the underlying system, rather than merely fitting the measurements. The numerical results show that our algorithms yield smaller models in less time, when compared to vector fitting.
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