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Enforcing Passivity of Macromodels via Spectral Perturbation of Hamiltonian Matrices
| Content Provider | Semantic Scholar |
|---|---|
| Author | Grivet-Talocia, Stefano |
| Copyright Year | 2003 |
| Abstract | This paper presents a new technique for the enforcement of passivity of linear macromodels. The proposed algorithm is applicable to state-space realizations of the macromodels in case the input-output transfer function is in admittance, impedance, hybrid, or scattering form. The core of the algorithm is the application of first-order perturbation theory to the eigenvalues of an associated Hamiltonian matrix. This allows both a precise definition of the frequency bands where passivity violations occur, and the determination of a new set of state matrices leading to passivity compensation. The main algorithm is very efficient in the case of small passivity violations, so that first-order perturbation is feasible. An application to passive macromodeling of a package structure is presented. Introduction The research that motivates this work is focused on the generation of linear lumped macromodels for multiport interconnects. Such macromodels are of paramount importance for the analysis and design of any high-speed electronic system. In fact, the Signal Integrity (SI) of such systems can only be assessed by accurate system-level simulations including suitable models for all the parts of the system that have some influence on the signals. Each of these models must be passive, otherwise serious instabilities may occur during the simulations. The standard procedure for the generation of lumped macromodels is to derive some rational approximation of the transfer matrix for the structure under investigation. Some techniques are available for the direct generation of passive macromodels (see, e.g., [4, 8]). These techniques require a circuit description (possibly including transmission lines) of the structure. However, such description is not always available. Therefore, alternative approaches have been proposed for the identification of lumped macromodels starting from input-output port responses, either in time or frequency domain (see, e.g., [5, 6]). Although very accurate approximations can be generated even for quite complex structures, the typical outcome of such methods is are stable but possibly non-passive models. This paper intends to propose a technique for the detection, characterization, and compensation of such passivity violations. Passivity may be defined in a loose sense as the inability of a given structure to generate energy. The precise definition of passivity [1] requires that the transfer matrix under investigation be positive real (in case of hybrid representations of the multiport) or bounded real (in case of scattering representations). The direct application of these definitions for testing passivity, however, requires a frequency sweep since these conditions need to be checked at any frequency. The results of such tests, therefore, depend on an accurate sampling of the frequency axis, which is not a trivial task. Erroneous results may occur. For this reason, purely algebraic passivity tests are highly desirable. Fortunately, the Positive Real Lemma and the Bounded Real Lemma provide an answer to this problem [3]. These results provide a connection between the passivity definitions and various equivalent algebraic conditions. These conditions can be expressed via feasibility of Linear Matrix Inequalities (LMI), or via existence of solutions to equivalent Algebraic Riccati Equations (ARE), or via the spectral properties of associated Hamiltonian matrices. For an excellent review and for a rich bibliography on the subject we refer the reader to [3]. In this work, we focus on the latter formulation using Hamiltonian matrices, since a study of their spectral properties leads not only to a precise criterion for passivity check, but also to a simple algorithm for the passivity compensation in case some passivity violations are detected. Both the passivity check and the compensation algorithms proposed in this paper are based on first-order spectral perturbations of the associated Hamitonian matrices. Preliminary and Notations In this work, we concentrate on the characterization and compensation of passivity of a given linear macromodel. Our starting point will be a state-space realization of the macromodel, ragardless of the particular macromodeling algorithm that was used to derive it. Therefore, we consider a linear time-invariant multiport system M in state space form |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://iris.polito.it/retrieve/handle/11583/1412827/46364/cnf-2003-spi-Hamiltonian.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
